# Flattening the Curve of Flexible Space Robotics [PERSON]
## 2 Materials and Methods
### Modeling and Analysis of Flexible Space Robots
#### 2.1.2 Flexible Body Dynamics  
Discretizing the flexible appendages into a chosen number of nodes (four nodes used here illustratively, as depicted in Figure 7b) and rotational and translational equations from Section 2.1.1 are repeatedly applied at each of the chosen nodes resulting in a system of equations of dimension equal to the chosen number of nodes. The system of equations expresses in three axes is parameterized in accordance with the single (\(\hat{z}\)-axis) rotation depicted in Figure 8a, representing the flexible space robot simulator depicted in Figure 8b. Notice the end effector manipulator position in this illustration is achieved by control of the attitude of the central rigid body. Omitted here also is the grasping action of the robotic gripper (depicted in Figure 7b). The rigid spacecraft hub in Figure 8 is parameterized in the reference frame (black font \(\hat{x},\hat{y},\hat{z}\)) centered at point frame \(O\), which rotates an angle \(\theta\) to angular positions designated by the red dashed line with a red font \((\hat{X},\hat{Y},\hat{Z})\), while the actuator wheel (purple font \(\hat{x},\hat{y},\hat{z}\)) is centered at point frame \(O_{W}\), which rotates an angle \(\theta_{W}\) to angular positions designated by the purple dashed lines with a purple font \((\hat{x},\hat{y},\hat{z},\hat{z}\). The vector of multi-dimensional forces \(F\) on the robotic arm are decomposed into radial \(\mathbb{F}_{F}\) and normal \(\overline{\mathbb{u}}_{F}\) components. Equations of Motion for the flexible system (rigid body plus flexible appendages may be conveniently derived using Euler-Lagrange's Method. [PERSON]'s method is appropriate regardless of various coordinate systems since it utilizes the systems kinetic and potential energy (V) in Equation (13) to derive the equations of motion. \[\text{{{Kinetic} energy}}=\frac{1}{2}=\int_{R}(V_{R}\cdot V_{R})dm+\int_{F}(V_{ F}\cdot V_{F})dm+\int_{W}(V_{W}\cdot V_{w})dm \tag{13}\]  
where:  
\(V_{R}=\hat{\theta}\hat{k}\times r_{R}\) is the velocity of a particle on the rigid body;  
\(V_{F}=\hat{\theta}\hat{k}\times r_{F}+\hat{\theta}\hat{k}\times\hat{\delta}\) is the velocity of a particle on the flexible body;  
\(V_{W}=\left(\hat{\theta}+\hat{\theta}_{W}\right)\hat{k}\times r_{W}+\hat{ \theta}\hat{k}\times r_{o}\) is velocity of a reaction wheel particle;  
\(r\) vectors are position vectors to the respective particles;  
\(\delta\) is elastic deformation vectors of particles on the flexible body;  
\(\hat{i},\hat{j},\hat{k}\) and \(\hat{k}\) are unit vectors in the \(x^{\prime},y^{\prime},z^{\prime}\) axes of the reaction wheel. Figure 8: Schematic representation (**a**) of the flexible space robot depicted in Figure 7, representing the free-floating flexible robot simulator at the U.S. Naval Postgraduate School depicted in (**b**). Expanding all the terms in the equation for kinetic energy produces Equation (14):  
\[ \begin{array}{ll}\mbox{{\it Kinetic energy}}&=\frac{1}{2}\int\limits_{R} \left(x_{R}^{2}+y_{R}^{2}\right)dm+\frac{1}{2}\int\limits_{F}\left[\dot{\theta} ^{2}\left(x_{R}^{2}+y_{R}^{2}\right)+\left(\dot{\delta}_{x}^{2}+\dot{\delta}_{ y}^{2}\right)+\dot{\theta}^{2}\left(\delta_{x}^{2}+\delta_{y}^{2}\right) \right]dm\\ &+\frac{1}{2}\int\limits_{F}\left[\dot{\theta}^{2}\left(x_{0}^{2}+y_{0}^{2} \right)+\dot{\theta}^{2}\left(x_{w}^{2}+y_{w}^{2}\right)+2\dot{\theta}(x_{0}x_ {w}+y_{0}y_{w})+\dot{\theta}_{w}^{2}\left(x_{w}^{2}+y_{w}^{2}\right)\right]dm \\ &\qquad\qquad+\frac{1}{2}\int\limits_{W}\left[2\dot{\theta}\dot{\theta}_{w} \left(x_{w}^{2}+y_{w}^{2}\right)+2\dot{\theta}_{w}\dot{\theta}(x_{0}x_{w}+y_{ 0}y_{w})\right]dm\end{array}  \tag{14}\]  
Utilizing the generalized coordinates \(\theta,\theta_{W},\delta\), assuming the reaction wheel rotates about its center of mass, and invoking the small displacement assumption, simplifies the kinetic energy equation to Equation (15):  
\[