## 4 GNC/FDIR System Design and Analysis
### Guidance and Passive Safety  
In the application of Active Debris Removal, it is usually desirable to have an autonomous guidance system that can run independently of ground intervention. This guidance, being autonomous, is defined and validated prior to the flight. In CS-1, the sanity of the guidance approach was performed by assessing the controllability of the approach trajectory. In particular, the sequence of operation is depicted in Figure 10 and can be summarized as the following:  
* A station keeping at FKP, that is placed on the V-bar;
* Forced motion towards IPP;
* Station keeping at IPP;
* Forced motion towards the V-bar, to a point at the same distance of IPP;
* Station keeping;
* Forced motion until Capture. Figure 10: Close Rendezvous and Capture nominal trajectory in LVLH. From the analysis of the relative position and velocity profiles in the LVLH frame, the required forces can be determined [31]. Namely, the time evolution of the position and velocity in LVLH coordinates can be translated into a thruster command force as described in [31]. Regarding attitude, target pointing defines the angular position and velocity as functions of the linear position and velocity. Therefore, the time evolution of angular coordinates can be computed as a function of linear coordinates. Finally, the required torques are obtained from the attitude and angular velocity using [PERSON]'s rotation equations [33]. These are illustrated in Figure 11. Note that this yields a nominal profile for the forces and torques, and hence an authority margin should be considered in order to compensate for errors and dispersions. The nominal profile is within the maximum control authority, with an 80% margin. This defines a necessary condition for the servicer to control its position and attitude pointing during the mission. If the maximum control authority is exceeded within the KOZ, then an Abort or Cancel are commanded, as defined in Section 3.4; if this happens outside the KOZ, then a Cancel is initiated towards a recovery point, usually FKP. Another important aspect defined in [11] is that, prior to any rendezvous and/or capture operations, the servicer is capable of performing an inspection of the capture point of the client. This is satisfied by the design of the approach trajectories, described by ROEs [36]. For the sake of clarity, Relative Orbital Elements are defined as follows:  
\[\delta\alpha= \begin{pmatrix}\delta a\\ \delta\lambda\\ \delta e_{x}\\ \delta e_{y}\\ \delta i_{x}\\ \delta i_{y}\end{pmatrix} = \begin{pmatrix}(a_{c}-a)/a\\ (u_{c}-u)+(\Omega_{c}-\Omega)\cos i\\ e_{x_{c}}-e_{x}\\ e_{y_{c}}-e_{y}\\ i_{c}-i\\ (\Omega_{c}-\Omega)\sin i\end{pmatrix} \]  
and are based on the well-known Keplerian (absolute) orbital elements as follows  
* _a_: semi-major axis,
* _e_: eccentricity,
* _i_: inclination,
* _\(\Omega\)_: right ascension of the ascending node,
* _\(\omega\)_: argument of periapsis,  
Figure 11: Actuation required to perform the forced motion of the rendezvous profile: force (**a**) and torque (**b**) profiles. The lines show the body axes where the quantities are applied vs. time. \(\theta\): true anomaly,
* \(M\): mean anomaly. Note that subscript \(c\) denotes the chaser, or servicer, and no subscript denotes the target elements. Moreover, \(u=\omega+M\) is the mean argument of latitude. Finally, \(e_{x}\) and \(e_{y}\) are defined as \(e\mathrm{cos}\,\omega\) and \(e\mathrm{sin}\,\omega\), respectively. An example of trajectory, defined through ROEs, is depicted in Figure 12, assuming a circular target orbit and \(\delta a=0\), which results in a closed ellipse centered in the V-bar. The notation in Figure 12 assumes  
\[\delta e= \begin{pmatrix}\delta e_{x}\\ \delta e_{y}\end{pmatrix} ,\ \delta i= \begin{pmatrix}\delta i_{x}\\ \delta i_{y}\end{pmatrix} .\]  
In particular, inspection is performed with a fly-around trajectory with a constant ROE.