## 3 Spacecraft Dynamics
### Governing Equations of Smart Structures  
This section is dedicated to the introduction of the governing equation of a smart active structure and to the modeling approach used to cope with the uncertainties that appear in the knowledge of some parameters of the dynamic model. Generally, the system state vector can be defined as follows  
\[u=[x_{G},\theta,\eta] \tag{10}\]  
where \(x_{G}\) represents the position of the platform's center of gravity (G) relative to an inertial frame (Figure 8), \(\theta\) the attitude of the body reference frame in relation to the inertial system, and \(\eta\) denotes the modal amplitudes of a flexible appendage attached to a central satellite platform. Therefore, the dynamic system of the general electro-elastic piezo-structure, which considers the mechanical properties of both passive and active elements, is described by the following set of equations:  
\[M_{u}^{G}\ddot{u}+\mathbf{C_{u}}\dot{u}+\mathbf{K_{u}}u+N_{l}=f^{G}+\mathbf{U_{a}}\varphi_ {a}+\mathbf{U_{s}}\varphi_{s} \tag{11}\]  
where \(M_{u}^{G}\) is the total mass matrix of the system, \(\mathbf{C_{u}}\) and \(\mathbf{K_{u}}\) are the damping and stiffness matrices, \(N_{l}\) contains the non-linear dynamics terms and \(f^{G}\) the generalized forces exerted  
Figure 8: Spacecraft model. Figure 9: Spacecraft free-free modes: (**a**) 1 st mode at 0.36 Hz, symmetric bending; (**b**) 2 nd mode at 0.52 Hz, asymmetric bending; (**c**) 3 rd mode at 1.29 Hz, torsion. Spectrum values illustrate the magnitude of normal modes eigenvectors, which are generally defined up to an arbitrary scale factor. on the system CoG (forces \(f_{G}\), torques \(\tau_{G}\), and projection of forces \(\vec{f}_{G}\) on the modal base), \(\mathbf{U_{a}}\) and \(\mathbf{U_{s}}\) are the input matrices of the piezo actuators and sensors, respectively, while \(\varphi_{a}\) and \(\varphi_{s}\) are the vectors including the voltages associated to the smart devices. The extended expression of the terms \(N_{l}\) can be found in [33] and will not be presented here for the sake of brevity. In particular, the total mass matrix \(\mathbf{M}_{u}^{G}\) can be written as  
\[\mathbf{M}_{u}^{G}= \begin{bmatrix}\mathbf{M}_{t}&\sum_{i}^{N}\vec{\mathbf{p}}_{O}^{A_{i} \times}&\mathbf{L}_{k}^{A_{1}}&\cdots&\mathbf{L}_{k}^{A_{N}}\\ \sum_{i}^{N}\vec{\mathbf{p}}_{O}^{A_{i}\times^{T}}&\mathbf{J}_{O}^{A_{i}}&\mathbf{S}_{k}^{ A_{1}}&\cdots&\mathbf{S}_{k}^{A_{N}}\\ \mathbf{L}_{k}^{A_{1}}{}^{T}&\mathbf{S}_{k}^{A_{1}}{}^{T}&\mathbf{I_{1}}&\cdots&\mathbf{0}\\ \vdots&\vdots&\mathbf{0}&\ddots&\mathbf{0}\\ \mathbf{L}_{k}^{A_{N}}{}^{T}&\mathbf{S}_{k}^{A_{N}}{}^{T}&\mathbf{0}&\cdots&\mathbf{I}_{N}\\ \end{bmatrix}  \tag{12}\]  
where \(\mathbf{M}_{t}=\left(\sum_{i}^{N}M_{A_{i}}\right)+\mathbf{M}_{b}+\left(\sum_{i}^{N_{a} }M_{p_{i}}^{\mathbf{a}}\right)+\left(\sum_{i}^{N_{a}}M_{p_{i}}^{\mathbf{s}}\right)\) total mass of the active flexible spacecraft, with \(\mathbf{M}_{A_{i}}\) mass matrices of the \(i\)-th appendages, \(\mathbf{M}_{b}\) mass of the satellite platform, \(\mathbf{M}_{p}^{a}=b_{a}^{T}b_{a}m_{a}\) and \(\mathbf{M}_{p}^{a}=b_{s}^{T}b_{s}m_{a}\) mass matrices of the piezo actuators and sensors, respectively, with \(b_{a},m_{a},N_{a}\) and \(b_{s},m_{s},N_{s}\) influence vectors of the smart devices on the structure degrees of freedom [34], mass and number of the actuators and sensors respectively, \(N\) total number of flexible appendages. The \(\mathbf{M}_{u}^{G}\) matrix also includes the following terms: \(\vec{\mathbf{p}}_{G}^{A_{i}\times}\) skew matrix containing the static moment of the system, with respect to the spacecraft center of gravity G in the body reference frame, and \(\mathbf{J}_{G}^{A_{i}}\) total moment of inertia of the system with respect to G in the body frame. The variables \(\mathbf{L}_{k}^{A_{i}}\) and \(\mathbf{S}_{k}^{A_{i}}\) include the translation and rotation modal participation factors (coupling with the rigid motion), respectively, while \(\mathbf{I}_{k}\) is the identity matrix (structural modes are normalized with respect to mass).