Read Richard Langley’s introduction to this article: “Innovation Insights: Falcon Gold analysis redux”
Figure 1: Diagram of cis-lunar space, which includes the real GPS sidelobe data collected on an HEO space vehicle. (All figures provided by the authors)
As part of NASA’s increased interest in returning to the moon, the ability to acquire accurate, onboard navigation solutions will be indispensable for autonomous operations in cis-lunar space (see Figure 1). Artemis I recently made its weeks-long journey to the Moon, and spacecraft carrying components of the Lunar Gateway and Human Landing System are planned to follow suit. During launch and within the GNSS space service volume, space vehicles can depend on the robust navigation signals transmitted by GNSS constellations (GPS, GLONASS, BeiDou, and Galileo). However, beyond this region, NASA’s Deep Space Network (DSN) serves as the system to track and guide lunar spacecraft through the dark regions of cis-lunar space. Increasingly, development of a lunar navigation satellite system (LNSS) that relies on a low size, weight and power (SWaP) “smallSat” constellation is being discussed for various possible orbits such as low lunar orbit (LLO), near rectilinear halo orbit (NRHO) and elliptical frozen orbit (ELFO).
Figure 2: DPE 3D (left) and 2D (right) spatial correlogram shown on a 3D north-east grid.
We have implemented direct positioning estimation (or collective detection) techniques to make the most of the limited and weak GPS signals (see Figure 2) that have been employed in other GNSS-degraded environments such as urban canyons. The algorithm used in conventional GNSS positioning employs a two-step method. In the first step, the receiver acquires signals to get a coarse estimate of the received signal’s phase offset. In the second step, the receiver tracks the signals using a delay lock loop coupled with a phase or frequency lock loop. The second step enables the receiver to get fine measurements, ultimately used to obtain a navigation solution. In the scenario addressed in our work, where a vehicle is navigating beyond the GPS satellite constellation, the signals are weak and sparse, and a conventional GPS receiver may not be able to acquire or maintain a lock on a satellite’s sidelobe signals to form a position solution. For a well-parameterized region of interest (that is, having a priori knowledge of the vehicle orbital state through dynamic filtering), and if the user’s clock error is known within a microsecond, a direct positioning estimator (DPE) can be used to improve acquisition sensitivity and obtain better position solutions. DPE works by incorporating code/carrier tracking loops and navigation solutions into a single step. It uses a priori information about the GPS satellites, user location, and clocks to directly estimate a position solution from the received signal. The delay-Doppler correlograms are first computed individually for the satellites and are then mapped onto a grid of possible candidate locations to produce a multi-dimensional spatial correlogram. By combining all signals using a cost function to determine the spatial location with the most correlation between satellites, the user position can be determined. As mentioned, signals received beyond the constellation will be sparse and weak, which makes DPE a desirable positioning method.
BACKGROUND
The proposed techniques draw from several studies exploring the use of weak signals and provide a groundwork for developing robust direct positioning methods for navigating beyond the constellation. NASA has supported and conducted several of the studies in developing further research into the use of signals in this space.
A study done by Kar-Ming Cheung and his colleagues at the Jet Propulsion Laboratory propagates the orbits of satellites in GPS, Galileo, and GLONASS constellations, and simulates the “weak GPS” real-time positioning and timing performances at lunar distance. The authors simulated an NRHO lunar vehicle based on the assumption that the lunar vehicle is in view of a GNSS satellite as long as it falls within the 40-degree beamwidth of the satellite’s antenna. The authors also simulate the 3D positioning performance as a function of the satellites’ ephemeris and pseudorange errors. Preliminary results showed that the lunar vehicle can see five to 13 satellites and achieve a 3D positioning error (one-sigma) of 200 to 300 meters based on reasonable ephemeris and pseudorange error assumptions. The authors also considered using relative positioning to mitigate the GNSS satellites’ ephemeris biases. Our work differs from this study in several key ways, including using real data collected beyond the GNSS constellations and investigating the method of direct positioning estimation for sparse signals.
Luke Winternitz and colleagues at the Goddard Space Flight Center described and predicted the performance of a conceptual autonomous GPS-based navigation system for NASA’s planned Lunar Gateway. The system was based on the flight-proven Magnetospheric Multiscale (MMS) GPS navigation system augmented with an Earth-pointed high-gain antenna, and optionally, an atomic clock. The authors used high-fidelity simulations calibrated against MMS flight data, making use of GPS transmitter patterns from the GPS Antenna Characterization Experiment project to predict the system’s performance in the Gateway NRHO. The results indicated that GPS can provide an autonomous, real-time navigation capability with comparable, or superior, performance to a ground-based DSN approach using eight hours of tracking data per day.
In direct positioning or collective detection research, Penina Axelrad and her colleagues at the University of Colorado at Boulder and the Charles Stark Draper Laboratory explored the use of GPS for autonomous orbit determination in geostationary orbit (GEO). They developed a novel approach for directly detecting and estimating the position of a GEO satellite using a very short duration GPS observation period that had been presented and demonstrated using a hardware simulator, radio-frequency sampling receiver, and MATLAB processing.
Ultimately, these studies and more have directed our research in exploring novel methods for navigating beyond the constellation space.
DATA COLLECTION
The data we used was collected as part of the U.S. Air Force Academy-sponsored Falcon Gold experiment and the data was also post-processed by analysts from the Aerospace Corporation. A few of the key notions behind the design of the experiment was to place an emphasis on off-the-shelf hardware components. The antenna used on board the spacecraft was a 2-inch patch antenna and the power source was a group of 30 NiMH batteries. To save power, the spacecraft collected 40-millisecond snapshots of data and only took data every five minutes. The GPS L1 frequency was down-converted to a 308.88 kHz intermediate frequency and was sampled at a low rate of 2 MHz (below the Nyquist rate) and the samples were only 1- bit wide. Again, the processing was designed to minimize power requirements.
METHODS AND SIMULATIONS
To test our techniques, we used real data collected from the Falcon Gold experiment on a launch vehicle upper stage (we’ll call it the Falcon Gold satellite) which collected data above the constellation on a HEO orbit. The data collected was sparse, and the signals were weak. However, the correlation process has shown that the collected data contained satellite pseudorandom noise codes (PRNs). Through preliminary investigation, we find that the acquired Doppler frequency offset matches the predicted orbit of the satellite when propagated forward from an initial state. The predicted orbit of the satellite was derived from the orbital parameters estimated using a batch least-squares fit of range-rate measurements using Aerospace Corporation’s TRACE orbit-determination software. The propagation method uses a Dormand-Prince eighth-order integration method with a 70-degree, first-order spherical harmonic gravity model and accounting for the gravitation of the Moon and Sun. The specifics of this investigation are detailed below.
Figure 3: GPS constellation “birdcage” (grey tracks), with regions of visibility near the GPS antenna boresight in blue and green for the given line-of-sight from the Falcon Gold satellite along its orbit (orange).
The positions of the GPS satellites are calculated using broadcast messages (combined into so-called BRDC files) and International GNSS Service (IGS) precise orbit data products (SP3 files). GPS satellites broadcast signals containing their orbit details and timing information with respect to an atomic clock. Legacy GPS signals broadcast messages contain 15 ephemeris parameters, with new parameters provided every two hours. The IGS supports a global network of more than 500 ground stations, whose data is used to precisely determine the orbit (position and velocity in an Earth-based coordinate system) and clock corrections for each GNSS satellite. These satellite positions, along with the one calculated for the Falcon Gold satellite, allowed for the simulation of visibility conditions. In other words, by determining points along the Falcon Gold satellite trajectory, we determine whether the vehicle will be within the 50° beamwidth of a GPS satellite that is not blocked by Earth.
Figure 3 shows a plot rendering of the visibility conditions of the Falcon Gold satellite at a location along its orbit to the GPS satellite tracks. Figure 4 depicts three of the 12 segments where signals were detected and compares the predicted visibility to the satellites that were actually detected. A GPS satellite is predicted to be visible to the Falcon Gold satellite if the direct line-of-sight (DLOS) is not occluded by Earth and if the DLOS is within 25° of the GPS antenna boresight (see Figure 5).
Figure 4: Predicted visibility of direct line-of-sight to each GPS satellite where a blue line indicates the PRN is predicted to be visible but undetected. A green line is predicted to be visible and was detected, and a red line indicates that the satellite is predicted to not be visible, but was still detected.
Figure 5: Depiction of the regions of a GPS orbit where the Falcon Gold satellite could potentially detect GPS signals based on visibility.
As a preliminary step to evaluate the Falcon Gold data, we analyzed the Doppler shifts that were detected at 12 locations along the Falcon Gold trajectory above the constellation. By comparing the Doppler frequency shifts detected to the ones predicted by calculating the rate of change of the range between the GPS satellites and modeled Falcon Gold satellite, we calculated the range rate root-mean-square error (RMSE). Through this analysis, we were able to verify the locations on the predicted trajectory that closely matched the detected Doppler shifts.
These results are used to direct our investigations to regions of the dataset to parameterize our orbit track in a way to effectively search our delay and Doppler correlograms to populate our spatial correlograms within the DPE. Figure 6 shows the time history of the difference of predicted range rates on the trajectory and the detected range rates on the trajectory. That is, a constant detected range rate value is subtracted from a changing range rate for the duration of the trajectory and not just at the location on the trajectory at the detect time (dashed vertical line). From this we can see that the TRACE method gives range rates near the detected ranges at the approximate detection time for the 12 different segments.
Figure 6: Plots depicting the 12 segments of detection and the corresponding time history of differences of range-rate values for each GPS PRN detected. The time history is of the range-rate difference between the predicted range rate from the TRACE-estimated trajectory and the constant detected range rate at the detection time (vertical line).
Excluding Segment 12, which was below the MEO constellation altitude, Segment 6 has more detected range rates than that of the other segments. On closer inspection of this segment, and using IGS precise orbit data products, it appears that the minimum RMSE of the range rates from the detected PRNs is off from the reported detection time by several seconds (see Figure 7). Investigating regions along the Falcon Gold TRACE-estimated trajectory and assuming a mismatch in time tagging results in a location (in Earth-centered Earth-fixed coordinates) with a lower RMSE for the predicted range rates compared to detected range rates.
Figure 7: Range-rate difference between the predicted range rate from the TRACE-estimated trajectory and the constant detected range rate at the detection time (left). A portion of the trajectory around Segment 6 with the TRACE-estimated location at the time of detection (red) and the location with the minimum RMSE of range rate (black).
To determine the search space for the DPE, we first determine the location along the original TRACE-estimated trajectory with the minimum RMSE of range rates for each segment. Then we propagate the state (position and velocity) at the minimum location to the Segment 6 time stamp. If the time segment has more than three observed range rates (Segment 6 and Segment 12), we perform a least squares velocity estimate using the range-rate measurements, using the locations along the trajectory and selecting the location with the smallest RMSE. Then, for Segment 12, the position and velocity obtained from least squares is propagated backwards in time to the Segment 6 timestamp. All of these points along the trajectory as well as the original point from the TRACE estimated trajectory are used in a way similar to the method of using a sigma point filter. Specifically, the mean and covariance of the position and velocity values are used to sample a Gaussian distribution. This distribution will serve as the first iteration of the candidate locations for DPE. There were a total of three iteration steps and at each iteration the range of clock bias values over which to search was refined from a spacing of 1,000 meters, 100 meters, and 10 meters. Also on the third iteration, the sampled Gaussian distribution was resampled with 1,000 times the covariance matrix values in the directions perpendicular to the direction to Earth. This was done to gain better insight into the GPS satellites that were contributing to the DPE solution.
RESULTS
Figure 8 shows the correlation peaks for each of the signals reported to be detected using a 15-millisecond non-coherent integration time within the DPE acquisition. Satellite PRNs 4, 16 and 19 are clearly detected. Satellite PRN 29 is less obviously detected, but the maximum correlation value is associated with the reported detected frequency. However, this is the peak detected frequency only if the Doppler search band is narrowly selected around the reported detected frequency. Similarly, while the peak code delay shows a clear acquisition peak for PRNs 4, 16 and 19, for PRN 29 the peak value for code delay is more ambiguous with many peaks of similar magnitude of correlation power. Figure 8 depicts the regions around the max peak correlation chip delay.
Figure 8: Acquisition peak in frequency (left) and time (right) for PRN 4, 16, 19 and 29. The correlograms are centered on the frequency predicted from the range rate calculated along the trajectory.
For the first iteration of DPE, the peak coordinated acquisition values for PRN 16 and PRN 4 are chosen for the solution space. From the corresponding spatial correlogram, the chosen candidate solution is roughly 44 kilometers away from the original position estimated using TRACE.
For the second iteration of DPE, the clock bias is refined to search over a 100-meter spacing. The peak values for PRN 16 and PRN 19 are chosen for the solution space and the chosen candidate solution is roughly 38 kilometers away from the original position estimated using TRACE.
For the final iteration, Figures 9 and 10 depict the solutions with the 10-meter clock bias spacing and the approach of spreading the search space over the dimension perpendicular to the direction of Earth. Again, this was done to illustrate how the peak correlations appear to be drawing close to a single intersection location. However, the results fall short of the type of results shown in the spatial correlogram previously depicted in Figure 2 when many satellite signals were detected.
Figure 9: Acquisition peaks plotted in the time domain with the candidate location chosen at the location of the vertical black line for the detected PRNs for the third iteration of the DPE method.
Figure 10: Spatial correlogram with the candidate location chosen at the location of the black circle for the detected PRNs for the third iteration of DPE method. The original TRACE-estimated position is indicated by a red circle. The two positions are approximately 28 kilometers apart.
A similar iterative method was followed using not just the four detected PRNs, but any satellite that was predicted to be visible with the relaxed criteria allowing for visibility based on receiving signals from the first and second sidelobes of the antenna. This is predicted using a larger 40° away from the GPS antenna boresight criterion. The final spatial correlogram (Figure 11) shows similar results to the intersections shown in Figure 10. However, there is potentially another PRN shown with a peak contribution near the original intersection point. These results are somewhat inconclusive and will need to be investigated further.
Figure 11: Spatial correlogram with the candidate location chosen at the location of the black circle for the detected PRNs for the third iteration of DPE method using additional satellites. The original TRACE-estimated position is indicated by a red circle. The two positions are approximately 24 kilometers apart.
CONCLUSIONS AND FUTURE WORK
Our research investigated the DPE approach of positioning beyond the GNSS constellations using real data. We will further investigate ways to parameterize our estimated orbit for use within a DPE algorithm in conjunction with other orbit determination techniques (such as filtering) as our results were promising but inconclusive. Some additional methods that may aid in this research include investigating the use of precise SP3 orbit files over the navigation message currently used (BRDC) within our DPE approach. Also, some additional work will need to be completed in determining the possibility of time tagging issues that could result in discrepancies and formulating additional methods related to visibility prediction that could aid in partitioning the search space. Additionally, we plan to investigate other segments where few signals were detected, but where more satellites are predicted to be visible (a better test of DPE). Finally, using full 40-millisecond data segments rather than the 15 milliseconds used to date may provide the additional signal strength needed to give more conclusive results.
ACKNOWLEDGMENTS
This article is based on the paper “Direct Positioning Estimation Beyond the Constellation Using Falcon Gold Data Collected on Highly Elliptical Orbit” presented at ION ITM 2023, the 2023 International Technical Meeting of the Institute of Navigation, Long Beach, California, January 23–26, 2023.
KIRSTEN STRANDJORD is an assistant professor in the Aerospace Engineering Department at the University of Minnesota. She received her Ph.D. in aerospace engineering sciences from the University of Colorado Boulder.
FAITH CORNISH is a graduate student in the Aerospace Engineering Department at the University of Minnesota.
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