# Novel 3U Stand-Alone CubeSat Architecture for Autonomous Near Earth Asteroid Fly-By
## 3 Autonomous Navigation
This is the case of optical line-of-sight navigation via star trackers, which exploits the directions of visible bodies (planets, moons, asteroids), together with their actual position from on-board stored ephemeris, to estimate the state of the spacecraft in deep-space [12; 17; 18; 19]. This technique has a clear geometrical limitation due to the relative positions between observer (CubeSat) and tracked objects, which can be excluded from the Field-of-View (FOV) due to the Sun disk and/or blinding. Each target, with different departure times and different shapes of the orbit, will have its customized tracking strategy, which will moreover depend on whether the trajectory is low-thrust with relatively fixed attitude, or ballistic with a flexible attitude, during which the CubeSat can orientate to track a particular celestial object. The required accuracy at the end of the cruise depends on target NEAs characteristics and CubeSat hardware constraints. First, to have a fully autonomous navigation system, the CubeSat cannot receive updated target ephemeris from ground and it has to rely on propagated ephemeris stored on-board prior to launch time; so the state of the NEA target at the fly-by time will have a propagated covariance matrix associated to it, which will depend on the Time-of-Flight (ToF)--the larger it is the larger is the propagated uncertainty (as a rule of thumb). Moreover, each NEA is characterized by different size, shape, composition, and albedo, which together influence its brightness; in general they are faint objects, which can become visible in the FOV of the instrument (star tracker in this case), only few days before the fly-by. The latest the asteroid appears in the FOV, the smaller will be the achievable displacement (from nominal encounter position) to compensate position uncertainties (for given \(\Delta V\) availability), obtainable with a Trajectory Correction Maneuver (TCM). Asteroid apparent brightness can be modeled as [20]:  
\[V=H+5\log(rd)-2.5\log[(1-G)e^{-3.33(tan(\frac{\pi}{2}))^{0.63}}+Ge^{-1.87(tan( \frac{\pi}{2}))^{1.22}}], \tag{1}\]  
where \(d\) is the relative distance NEA-observer, \(r\) the relative distance NEA-Sun, \(\alpha\) the phase illumination angle, defined as the angle between the observer-target and Sun-target directions, \(H\) the absolute magnitude, and \(G\) the slope parameter. Off-the-shelf micro star trackers have a minimum detectable visible magnitude below 7.0 [21]. Setting the visual magnitude (to 7.0), the maximum relative distance CubeSat-target for the observation can be quantified, as a function of phase angle and Sun-NEA distance. Then, the number of days before the fly-by the asteroid becomes visible can be obtained, by linearly approximating the close approach. This value, together with the \(\Delta V\) available budget, influences how much the spacecraft trajectory can be corrected, once the asteroid is visible. Merging everything together, it is possible to preliminary compute the required accuracy at the end of the cruise for a particular NEA target. Figure 1 reports this process for the asteroid (433) Eros, taken as a target example. In (a) the number of days prior to the fly-by the NEA becomes visible is evaluated as a function of phase angle and distance NEA-Sun. Due to its brightness, in the best illumination condition, it becomes visible in the FOV only 9 days before the fly-by. A more realistic number oscillates between 6 and 7 due to the real approach angle, which it should be zero at the closest approach time to have the full asteroid disk in the FOV of the payload; assuming 8 days, which corresponds to 10\({}^{\circ}\) if the fly-by occurs close to the target perihelion, it is possible to compute the displacement from the nominal encounter position as a function of the available impulsive \(\Delta V\). The achievable displacement with only one impulsive TCM of 15 m/s is around 10,000 km (b).