# Flattening the Curve of Flexible Space Robotics [PERSON]
## 2 Materials and Methods
### Modal System Identification  
Convert the elastic deformation \(\delta\) into cantilever modal coordinates [12] using \(\delta_{x}=\sum_{i=1}^{n}\phi_{i}^{x}q_{i}(t)\) and \(\delta_{y}=\sum_{i=1}^{n}\phi_{i}^{y}q_{i}(t)\). This relationship describes that the displacement at time t of a point at position x along the FSS is related to the mode shape \(\phi\) (x and y components) and the generalized coordinate or modal coordinate, \(q_{i}(t)\), a sinusoidal function. The mode shape is a time-independent characteristic of the flexible structure. Using this relationship, kinetic energy \(T\) in terms of the generalized coordinates is Equation (17). \[\tau=\frac{1}{2}I_{zz}\dot{\theta}^{2}+\frac{1}{2}I_{w}\dot{\theta}_{w}^{2}+ \frac{1}{2}I_{w}\dot{\theta}\dot{\theta}_{w}+\frac{1}{2}\int\limits_{F}\left( \sum_{i=1}^{n}\sum_{i=1}^{n}\left[\phi_{i}^{x}\dot{\theta}_{j}^{y}+\phi_{i}^{y }\phi_{j}^{y}\right]\dot{q}_{i}\dot{q}_{j}\right)dm+\dot{\theta}\int\limits_{F} \left(x_{f}\sum_{i=1}^{n}\phi_{i}^{y}\dot{q}_{i}-y_{f}\sum_{i=1}^{n}\phi_{i}^{x }\dot{q}_{i}-\right)dm \tag{17}\]  
\[\tau=\frac{1}{2}I_{zz}\dot{\theta}^{2}+\frac{1}{2}I_{w}\dot{\theta}_{w}^{2}+ \frac{1}{2}I_{w}\dot{\theta}_{w}+\frac{1}{2}\sum_{i=1}^{n}\dot{q}_{i}^{2}+ \dot{\theta}\sum_{i=1}^{n}D_{i}\dot{q}_{i}^{2} \tag{18}\]  
where rigid elastic coupling term \(D_{i}=\int_{F}\left(x_{f}\phi_{i}^{y}-y_{f}\phi_{i}^{x}\right)dm\). Potential energy comes from stiffness and can be expressed in generalized coordinates as \(V=\frac{1}{2}\sum_{i=1}^{n}\omega_{i}^{2}q_{i}^{2}\), where \(\omega_{i}\) is the natural frequency of the \(i\)th mode. The virtual work is given [11] as \(\delta\mathcal{W}=\sum T\delta(\theta+\theta_{w})-\sum T\delta\theta+T_{D} \delta\theta=\sum T\delta\theta_{w}+T_{D}\delta\theta\) where \(T_{D}\) is disturbance torques and \(\sum T\) is total torques (not to be confused with T = kinetic energy). Revisiting the Euler-Lagrange equation, substituting potential and kinetic energy yields the equations of motion in terms of the generalized coordinates. \[\sum\tau=I_{zz}\ddot{\theta}+\sum_{i=1}^{n}D_{i}\ddot{q}_{i} \tag{19}\]  
\[\ddot{q}_{i}+2\dot{\delta}\omega_{i}\dot{q}_{i}+\omega_{i}^{2}\dot{q}_{i}+D_{i} \ddot{\theta}=0 \tag{20}\]  
Notice [PERSON]'s law results in the identical equations of motion accounting for the stiffness between the adjacent nodes of the flexible robot appendage. [PERSON]'s Law: \(\sum F=ma\) applies at each node of the system, where the coordinates are defined in the hybrid-coordinate system [12]. \[m_{1}\ddot{x}_{1}=-k_{1}x_{1}+k_{2}(x_{2}-x_{1}) \tag{21}\]  
\[m_{2}\ddot{x}_{2}=-k_{2}(x_{2}-x_{1})+F \tag{22}\]  
\[\underbrace{\left[ \begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array} \right]}_{[M]}\underbrace{\left\{ \begin{array}{cc}\ddot{x} _{1}\\ \ddot{x}_{2}\end{array} \right\}}_{\ddot{x}}+\underbrace{\left[ \begin{array}{ cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}\end{array} \right]}_{[K]}\underbrace{\left\{ \begin{array}{cc}x_{ 1}\\ x_{2}\end{array} \right\}}_{x}=\underbrace{\left\{ \begin{array}{cc}0\\ 1\end{array} \right\}}_{\{F\}}F \tag{23}\]  
\([M]\) is the system's global mass matrix (to be derived by finite element analysis);  
\([K]\) is the system's global stiffness matrix (to be derived by finite element analysis);  
\(\{F\}\) is the force vector. Including structural damping results in \([M]\big{\{}\ddot{x}\big{\}}+[C]\big{\{}\dot{x}\big{\}}+[K]\big{\{}x\big{\}}=\{F\}\). The two degrees of freedom are x/y displacements and \(\theta\) rotations. x/y rotations are mutually exclusive since we are not allowing plane stress and plane strain; that is, the members are not allowed to lengthen/shorten (bending displacements only). So, for the horizontal members, the x\({}_{1}\) in the equation of Newton's Law corresponds to the Y direction bending displacements. Similarly, for vertical members, the \(x_{1}\) corresponds to x-direction bending displacements. The nodal degrees of freedom are constrained to zero at the attachment point of the flexible and rigid bodies (forcing the appendage to stay attached).