In one of our previous posts we demonstrated what kind of benefit a small NanoFEEP thruster can have in collision avoidance situations. We showed different scenarios how a satellite can increase the distance to potentially colliding objects using NanoFEEP. We showed that the thruster is able to perform such a manoeuvre and concluded that electrical propulsion systems can help to contribute to a sustainable space environment, e.g. by performing collision avoidance manoeuvres.

Today we want to pickup where we left off and discuss a more detailed analysis, where different kinds of uncertainties are considered. We named these imperfections that will occur in the real world and need to be considered when planning a manoeuvre. These are

- uncertainty in the thrust that is generated,
- the pointing accuracy of the satellite and
- the uncertainty in the actual position of the satellite.

On the one hand a thruster may generate a slightly different thrust than commanded. This may be in the order of percents (e.g. 1% -5%). On the other hand a thruster may not be pointed perfectly into the desired direction, which also results in a different trajectory than aimed for. And lastly, the initial uncertainty in the position of the satellite may cause a deviation from the envisaged manoeuvre goal. This is caused by imperfect measurements (e.g. noise within the data of the GNSS receiver) together with an imperfect modelling of the orbital mechanics in space (e.g. imperfect modelling within the orbital propagator). Following we will look at each of the uncertainties and discuss their impact on the manoeuvre design.

### Uncertainties in the thrust

The results for the 4-maneuver cases are shown in the following figure. We assumed

- a low thrust uncertainty of 1% (red) and
- a high uncertainty case of 5% (blue).

The thrust error is imprinted randomly (assumed normal distribution) on the magnitude of the thrust vector within our simulations. In the below figure we see that for the case with low thrust uncertainty (red) the standard error lies at about 20 m around the expected mean value, the maximum error at 50 m. In the case with high thrust uncertainty we can see that the curve is shifted by 150 m below the 1% case, which means the separation towards the nominal orbit was less efficient with the higher thrust uncertainty. Furthermore, as expected the standard error has increased as well and lies at around 100 m. However, for both cases the errors from the propulsion system uncertainty are almost negligible, when considering that a separation of more than 3 km is reached.

### Uncertainties in the pointing

For the imperfect pointing of the thruster toward the desired direction, we also created two cases:

- a low pointing error of 1440 arcsec (0.4 degrees) and
- a high pointing error of 7200 arcsec (2.0 degrees).

The results of both cases are shown below. The thrust uncertainty has been deactivated for these cases and the pointing is randomly (assumed normal distribution) altered for all three angles individually in the satellite centered coordinate system. For the low-error case the standard deviation is at 8 m, for the high-error case it is at 16 m. Here the pointing has an even lower impact than the thrust uncertainty.

### Uncertainties in the state of the object

Of course, there are not only uncertainties in the maneuver, but also in the state of the object. At the time when you start designing a collision avoidance maneuver, you only know your own satellite’s state with a certain accuracy. Depending on how old the orbit is you have at hand, this uncertainty might be higher or lower. Below, we show the result for propagating the same object using different co-variances, which can be considered typical for the age of the state given. The thrust and pointing uncertainties are deactivated. The older the state that is used for the design of the manoeuvre, the further it needs to be extrapolated into the future. The further it is extrapolated, the more uncertainty is added to the solution and the error envelope widens. This is reflected in the results below. The figure shows a big variety for states that are older than 1-day (blue, green and black dash-dotted lines). The resulting standard errors at the end are

- +/-240 m for the 6-hour case,
- +/-810 m for the 1-day case,
- +/-980 m for the 2-day case,
- +/- 1280 m for the 3-day case.

These numbers hint at a chance that the “real” trajectory might be within the lower and the upper bounds. In turn this means that such an error range needs to be regarded at design time of the manoeuvre. The position difference that must be achieved to be “on the save side” must always regard the uncertainties from the initial state vector, which depends on the age of the vector itself. The uncertainty of the initial state vector has the biggest impact within analysis.

### OKAPI’s conclusion

In this follow-up on the post we extended the analysis in order to regard different kinds of uncertainties that occur in the real world. In the scenarios we looked at:

**the uncertainties of the propulsion system’s thrust and pointing, which were**almost negligible**and**

**the uncertaintiy of the initial position and velocity (state vector), which can have an important impact on the manoeuvre dimension**

If you are interested in this topic, we would love to get in touch with you! Leave a comment or send an e-mail to: contact@okapiorbits.space. If you are interested in the validation of electrical propulsion systems and their validation reach out!