## 2 Sequential Modelling Strategy
### Spacecraft Dynamics with Structural Considerations
#### 2.4.2 Single Rigid-Body Dynamics  
The rigid-body assumption involves neglecting terms associated with changes in mass and mass moment of inertia. It assumes that the body-fixed coordinate system has its origin at the centre of mass. With these considerations, this section derives the translational and rotational motions in various reference frames to simplify the analysis.  
When considering a net capturing scenario as a typical rendezvous between two spacecraft, the Clohessy-Wiltshire equations of relative motion become applicable [24]. These equations are defined with respect to the Local-Vertical-Local-Horizontal (LVLH) O-_xyz_ frame and serve as the linearised version of the general relative orbital motion of the target's equations of motion with respect to the chaser. These are as follows:  
\[\ddot{\mathbf{r}}_{d}=\omega_{n}^{2}\begin{pmatrix}3\ x_{d}\\ 0\\ -z_{d}\end{pmatrix}+2\omega_{n}\begin{pmatrix}\dot{y}_{d}\\ -\dot{x}_{d}\\ 0\end{pmatrix}+\frac{1}{m_{d}}\sum_{i=1}^{N}\mathbf{F}_{c,i}, \tag{9}\]  
where \(\mathbf{F}_{c,i}\) is the \(i^{th}\) node impact force (with \(N\) total impacts), \(\omega_{n}\) is the constant orbital rate, \(m_{d}\) is the debris object mass and \(x_{d}\), \(y_{d}\) and \(z_{d}\) are the radial, along-track and cross-track components of the relative position vector \(\mathbf{r}_{d/0}\) with relative acceleration \(\ddot{\mathbf{r}}_{d}\).  
Figure 4: Representation of reference frames, incorporating the inertial frame \(\mathcal{N}\), the body-fixed and hinge frames for the target (\(\mathcal{B}\) and \(\mathcal{H}\)), and the chaser-attached \(\mathcal{O}\)-_xyz_ frame (\(\mathcal{L}\)). The respective centres of gravity for the main body, solar array and the entire spacecraft are labelled as \(B_{c}\), \(S_{c}\) and \(CG\) in the visualisation. The solar array arm or yoke is omitted for simplicity.  
Using the single rigid-body assumption, the rotational dynamics are given by  
\[\mathbf{I}_{\mathrm{d}}\dot{\mathbf{\omega}}_{\mathrm{d}}+\mathbf{\omega}_{\mathrm{d}} \times\mathbf{I}_{\mathrm{d}}\mathbf{\omega}_{\mathrm{d}}=\sum_{i}\mathbf{M}_{\mathrm{CG },i}, \tag{10}\]  
where \(\mathbf{I}_{\mathrm{d}}\) is the debris inertia matrix and \(\mathbf{\omega}_{\mathrm{d}}\) is the the rotation rate vector of the body \(\mathcal{B}\)-frame with regard to the inertial \(\mathcal{N}\)-frame (or \(\mathbf{\omega}_{B/\mathcal{N}}\)). Lastly, \(\mathbf{M}_{\mathrm{CG},i}\) refers to all the applied moments around the space debris centre of gravity (CG). As this study focuses on the influence of the net, all other disturbances (orbital, aerodynamic and solar radiation) are neglected.  
Lastly, a kinematic equation is required to relate the dynamic Euler equation and the physical orientation of the \(\mathcal{B}\)-frame with angles \(\theta_{X},\theta_{Y}\) and \(\theta_{Z}\). Thus, the body rotation rate \(\mathbf{\theta}_{\mathrm{d}}\) can be computed using \(\mathbf{\omega}_{\mathrm{d}}\) as:  
\[\mathbf{\dot{\theta}}_{\mathrm{d}}=\mathbf{C}(\theta_{X},\theta_{Y},\theta_{Z}) \mathbf{\omega}_{\mathrm{d}}+\frac{\omega_{n}}{\cos(\theta_{Y})}\begin{pmatrix} \sin(\theta_{Z})\\ \cos(\theta_{Y})\cos(\theta_{Z})\\ \sin(\theta_{Y})\sin(\theta_{Z})\end{pmatrix} \tag{11}\]  
where \(\mathbf{C}(\theta_{X},\theta_{Y},\theta_{Z})\) is the transformation matrix [18] and \(\mathbf{\dot{\theta}}_{d}\) is the derivative of the vector of Euler angles \(\theta_{X}\) to \(\theta_{Z}\) (summarised as the \(\mathbf{\theta}\)-state), corresponding to the pitch (\(\dot{\theta}_{X}\)), roll (\(\dot{\theta}_{Y}\)) and yaw (\(\dot{\theta}_{Z}\)) rates.