## 3 Spacecraft Dynamics
### Governing Equations of Smart Structures
Moreover, the matrices \(\mathbf{C}_{u}\) and \(\mathbf{K}_{u}\) are defined as  
\[\mathbf{C}_{u}= \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&2\mathbf{\Sigma}\mathbf{\Omega}\\ \end{bmatrix} ,\quad\mathbf{K}_{u}= \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&\mathbf{\Omega}^{2}\\ \end{bmatrix}  \tag{13}\]  
where \(\mathbf{\Omega}^{2}=\mathbf{\Phi}^{T}\Big{(}\mathbf{K}_{uu}+\left(\sum_{i}^{N_{a}}\mathbf{K }_{uu_{i}}^{a}\right)+\left(\sum_{i}^{N_{a}}\mathbf{K}_{uu_{i}}^{a}\right)\Big{)} \mathbf{\Phi}\) diagonal matrix containing all squared frequencies of the \(N_{p}\) appendages cantilevered to the satellite, and \(\mathbf{\Sigma}\) diagonal matrix including the \(k\)--the damping factor of the corresponding elastic modes. In addition, \(\mathbf{\Phi}\) is the matrix containing the structure modes, \(\mathbf{K}_{uu}\) the stiffness matrix of the passive structure, \(\mathbf{K}_{uu}^{a}=b_{a}^{T}b_{a}k_{a}\) and \(\mathbf{K}_{uu}^{a}=b_{s}^{T}b_{s}k_{s}\) the stiffness matrices of the actuators and sensors, respectively, while \(k_{a}\) and \(k_{s}\) are the stiffness parameters of the smart devices. The finite element model includes a total of three flexible modes for each appendage. Moreover, the input matrices \(\mathbf{U_{a}}\) and \(\mathbf{U_{s}}\) can be written as  
\[\mathbf{U_{a}}= \begin{bmatrix}\mathbf{0}_{n_{r}\times n_{r}}\\ \mathbf{\Phi}^{T}\mathbf{K}_{u\varphi}^{a}\\ \end{bmatrix} ;\mathbf{U_{s}}= \begin{bmatrix}\mathbf{0}_{n_{r}\times n_{r}}\\ \mathbf{\Phi}^{T}\mathbf{K}_{u\varphi}^{a}\\ \end{bmatrix}  \tag{14}\]  
where \(\mathbf{K}_{u\varphi}^{a}\) and \(\mathbf{K}_{u\varphi}^{s}\) are the global electro-mechanical coupling matrices of the actuators and sensors, respectively, and \(n_{r}\) is the number of rigid degrees of freedom of the spacecraft. To fully describe the active system, Equation (11) should be coupled to the following relations concerning the piezoelectric actuators (indicated with the apex/subscript \(a\)) and sensors (written with apex/subscript \(s\)) [34]  
\[\mathbf{K}_{u\varphi}^{a}u+\mathbf{K}_{\varphi\varphi}^{a}\varphi_{a}=g_{a} \tag{15}\]  
\[\mathbf{K}_{u\varphi}^{s}u+\mathbf{K}_{\varphi\varphi}^{s}\varphi_{s}=g_{s} \tag{16}\]with \(i=1,...,N_{a}\) index of the \(i\)-th actuator, and \(j=1,...,N_{s}\) index of the \(j\)-th sensor. The matrices \(\mathbf{K}_{u\varphi}^{a}\), \(\mathbf{K}_{\varphi\varphi}^{a}\),\(\mathbf{K}_{u\varphi}^{a}\), and \(\mathbf{K}_{\varphi\varphi}^{a}\), and the terms \(g_{a}\) and \(g_{s}\) will be introduced in the specific sections concerning smart devices, namely Sections 3.2.1 and 3.2.2. Let us now consider placing \(N_{a}\) actuators and \(N_{s}\) sensors on a flexible structure. In the considered application, the number of installed actuators and sensors is the same, and it yields \(N_{a}=N_{s}\). The devices are assumed to be implemented between two adjacent nodes \(m\), \(n\) of the structure, each node with six degrees of freedom \(V_{m,n}\) as follows:  
\[V_{m,n}=[u_{m},v_{m},z_{m},\alpha_{m},\beta_{m},\theta_{m},u_{n},v_{n},z_{n}, \alpha_{n},\beta_{n},\theta_{n}] \tag{17}\]  
where \(u\), \(v\), and \(z\) indicate the displacements along the local finite element \(x\), \(y\), and \(z\)-axis, respectively while \(\alpha\), \(\beta\), and \(\theta\) the rotations about the same axes (Figure 10).