Single-epoch least-squares solution

In order to test the achievable performance in terms of position determination, the WeakHEO receiver was tested in several representative portions of the considered MTO. Here, two relevant portions are reported, each of one-hour duration. The first portion starts at approximately 36 000 km of altitude (altitude of the Geostationary Earth Orbit) and a second portion which starts at approximately the average distance of the Moon from the centre of the Earth of 384 400 km, very close to the apogee of the MTO. The single-epoch least-squares 3D position error for the considered portion, which starts at GEO altitude, is illustrated in Figure 4:20, while Figure 4:21 shows the GDOP for the same portion. Figure 4:22 and Figure 4:23 show the same quantities for the portion that starts at Moon altitude. The worsening of the relative geometry between the receiver and the transmitters over time (i.e., with the altitude, as can be seen in Figure 4:4) is clearly visible at GEO altitude, strongly affecting the positioning accuracy. At Moon altitude, as expected, the positioning error is very high, almost reaching 14 km, mainly due to huge GDOP values larger than 1000, but also due to the processing of weaker signals and then to a stronger receiver noise, which results in less- accurate pseudorange measurements (see Figure 3:24).

Figure 4:20 Single-epoch least-squares 3D position error for the considered portion that starts at GEO altitude.

Figure 4:21 GDOP for the considered portion that starts at GEO altitude.

time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 G D O P 9 10 11 12 13 14 15 16 17

Figure 4:22 Single-epoch least-squares 3D position error for the considered portion that starts at Moon altitude.

Figure 4:23 GDOP for the considered portion that starts at Moon altitude.

Note that the WeakHEO tests have not been carried out over the full MTO duration as it lasts almost 5 days and at the time of our study the receiver platform did not include the algorithm required to acquire new signals or reacquire previously acquired signals. It is also important to keep in mind that the current

WeakHEO receiver is only a proof of concept and thus not the final device ready to operate in a lunar

mission, for which clearly more and longer tests will be required.

4.3 Conclusions

Following the previous feasibility studies of GNSS as a navigation system to reach the Moon, this chapter described the proof of concept of the GPS L1 C/A “WeakHEO” receiver for lunar mission, wholly developed in the last two years. After highlighting the characteristics of the GPS L1 C/A signals for the considered MTO, which were identified in the previous chapter, the requirements and constraints in the receiver design were defined. Afterwards, the general receiver architecture was described, providing a more detailed description of the acquisition, tracking and navigation modules. These modules were designed specifically for use with

time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 E C I 3 D p o si ti o n e rr o r (m ) 0 2000 4000 6000 8000 10000 12000 14000 LS Solution time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 G D O P 950 1000 1050 1100 1150 1200

the dynamic environment and signal conditions seen in high-altitude space applications, and the architecture presented is capable of performing acquisition, tracking, data synchronization and demodulation down to a level of 15 dB-Hz, confirming the theoretical analysis conducted in Chapter 3. This is verified on the hardware with tests using representative RF signals produced by a GNSS simulator. The computation of a navigation solution was possible in all the considered portions of the considered MTO, confirming that, as concluded in Chapter 3, 15 dB-Hz allows for the processing of at least the four strongest signals. The navigation performance when using a single-epoch least-squares estimator is coarse, as expected from the simulations carried out in Chapter 3, thus requiring a further filtering. Indeed, an orbital filter will be implemented and tested in different configurations, as described in the next chapters.

5 Orbital filter design and

architecture

As discussed in Chapter 3 and assessed in Chapter 4, although weak, GNSS signals from the side lobes of the GNSS transmitters antennas or from the spill-over of the main lobe can still be acquired and tracked successfully for high Earth orbits up to Moon altitude. As already mentioned, similar results were obtained in other studies such as [56], [76] and [90], thus confirming the feasibility of using GNSS as a navigation system to reach the Moon. At the same time, such studies as well as Chapter 3 and Chapter 4 have also highlighted how coarse a GNSS stand-alone non-filtered navigation solution at Moon altitude can be. Indeed, the higher the receiver is flying above the GNSS constellation, the weaker the GNSS received signals are (thus affecting the number of visible satellites and the pseudorange accuracy from the visible ones) and the larger the Geometric Dilution Of Precision (GDOP) is (resulting in lower positioning accuracy).

On another hand, a spacecraft is constrained to move along a certain trajectory by the orbital forces acting on it. If GNSS observations are filtered through an orbital dynamics mathematical model that is able to predict the observations themselves, the achievable navigation accuracy can be much higher. This kind of data fusion is commonly known as “orbital filter”, which fuses GNSS observations with the prediction of the space dynamics and generally leads to better solutions than what can be achieved by using a single-epoch least-squares estimator.

While several research papers, such as [2], [91], [92] and [93], have already described the use of an orbital filter for LEO, this chapter describes the implementation of a GNSS-based orbital filter specifically designed for lunar missions. A relevant characteristic of such a filter is that it makes use of an adapting tuning along the whole MTO, function of the GNSS measurements prediction, which as seen in Chapter 3, strongly varies depending on the relative position between the receiver and the GNSS satellites.

The analysis is carried out for two configurations of the filter: in the first configuration, denoted “position-

based”, the measurement inputs of the filter are the single-epoch least-squares GNSS positions and velocity,

while in the second configuration, denoted “range-based”, the measurement inputs are the raw GNSS observations, pseudoranges and pseudorange rates.

Section 5.1 introduces the estimation method used in the filter, while section 5.2 describes the position-

based and the range-based configurations. Sections 5.3 and 5.4 define respectively the state vector and the

measurement vector. Section 5.5 characterizes the spacecraft dynamics model implemented. Sections 5.6 and 5.7 describe the implemented observation functions and observation matrix. Section 5.8 characterizes the computation procedure of the state transition matrix and section 5.9 describes the adaptive filter tuning strategy that has been implemented and adopted.

The contents of this chapter were published in the journal papers [21] and [22].