# Space Robot Sensor Noise Amelioration Using Trajectory Shaping
## 2 Materials and Methods
### Equations of Motion  
The equations of motion of the flexible spacecraft system are derived using the Lagrange method. Lagrangian Mechanics requires kinetic and potential energies. See Table 1 for parameter definitions.  
The kinetic energy of the entire flexible spacecraft system was found by summing the kinetic energies of the rigid main body, flexible appendage, and reaction wheel. The kinetic energies are written in terms of the moments of inertia and the modal coordinates, as seen in Equation (1). \(D_{i}\) are the rigid-elastic coupling terms and are defined in Equation (2). The modal coordinates are expressed in terms of \(q\) and \(F\), which are defined in Equations (4) and (5) for a given beam \(i\).  
\[T=\frac{1}{2}{I_{zz}}\dot{\theta}^{2}+\frac{1}{2}{I_{w}}\dot{\theta}_{w}^{2}+{I _{w}}\dot{\theta}\dot{\theta}_{w}+\frac{1}{2}{\sum_{i=1}^{n}}\dot{q}_{i}+\dot{ \theta}{\sum_{i=1}^{n}}D_{i}\dot{q}_{i} \tag{1}\]  
The potential energy is shown in Equation (3), where \(\omega_{i}\) are the natural frequencies of the flexible spacecraft system for mode \(i\).  
\[D_{i}=\int{[x_{F}\dot{\theta}_{i}y}^{y}-y_{F}\dot{\phi}_{i}{}^{x}]dm \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Definition** & **Variable** & **Definition** & **Variable** & **Definition** \\ \hline \(T\) & Kinetic energy & \(T\) & Natural frequency of i-th mode & \(V\) & Potential energy \\ \(I_{zz}\) & Moment of inertia of rigid body & \(\dot{\phi}_{i}\) & Modal coordinates & \(W\) & Displacement \\ \(I_{w}\) & Moment of inertia of reaction wheel & \(x_{F}\) & Final position, x & \(Q\) & Sheer forces \\ \(\theta\) & Angle of flexible appendage & \(y_{F}\) & Final position, y & \(M\) & Moments \\ \(\theta_{w}\) & Angle of reaction wheel & \(n\) & Number of independent modes & \(L\) & Lagrangian \\ \(q_{i}\) & Modal coordinates & \(D_{i}\) & Elastic decoupling coefficients & \(T_{D}\) & Disturbance torque \\ \hline \hline \end{tabular}
\end{table}
Table 1: Flexible Spacecraft System Parameters.  
Figure 2: Flexible spacecraft system. (**a**) Atop a (planar) air bearing table, a free-floating space robot is autonomously controlled. [20] Imagery and photographs of the Department of Defense are in the public domain, unless otherwise noted [21]; (**b**) schematic diagram of the free-floating flexible space system depicted in subfigure (**a**). This schematic is identical to that used in references [10; 11], where this manuscript comprises the latest iteration of continuing research. No special permission is required to reuse all, or part of the article published by MDPI, including figures and tables [22].  
\[V=\frac{1}{2}\sum\limits_{i=1}^{n}{\omega_{i}^{2}{q_{i}}^{2}} \tag{3}\]  
\[\{q\}=\left\{W_{1}\quad\theta_{1}\quad W_{2}\quad\theta_{2}\ \right\}^{T} \tag{4}\]  
\[\{F\}=\left\{Q_{1}\quad M_{1}\quad Q_{2}\quad M_{2}\ \right\}^{T} \tag{5}\]  
The Lagrange equation, shown in Equation (6) is applied to the Lagrangian \(L\), where \(L=T-V\). The Lagrange method results in the equations of motion (EOM) of the flexible spacecraft system, which are shown in Equation (7).  
\[\frac{-d}{dt}\left(\frac{\delta L}{\delta\dot{u}_{i}}\right)-\left(\frac{ \delta L}{\delta u_{i}}\right)=Q_{i} \tag{6}\]  
\[EOM:\left\{\begin{aligned} & I_{zz}\ddot{\theta}+I_{w}\ddot{\theta}_{w}+ \sum\limits_{i=1}^{n}{D_{i}\dot{q}_{i}}=T_{D}\\ & I_{w}\left(\ddot{\theta}_{w}+\ddot{\theta}\right)=T\\ &\ddot{q}_{i}+{\omega_{i}}^{2}D_{i}\ddot{\theta}=0\end{aligned}\right. \tag{7}\]