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**DOP for LEO Navigation Constellation Design using Pseudorange and Doppler Shift**

*Brian McLemore and Mark L. Psiaki, Virginia Tech*

**Date/Time:** Thursday, Sep. 22, 9:20 a.m.

A study is conducted of Dilution of Precision (DOP) metrics for Low Earth Orbit (LEO) navigation constellation design when using a combination of carrier Doppler shift and pseudorange measurements to provide a point-solution capability. This study develops methods to analyze the beneficial effects of adding a sparse set of pseudorange-capable satellites to a large LEO constellation that provides a global navigation capability primarily through the use of carrier Doppler shift. It is hoped that the addition of pseudorange measurement capabilities in a small subset of the satellites will improve the otherwise poor timing accuracy of a Doppler-only system. The study is an extension of [1]. In [1], a Geometric Dilution of Precision (GDOP) analysis was developed to predict navigation accuracy of massive LEO constellations producing both pseudorange and carrier Doppler shift measurements. A single GDOP metric was not found that works for all cases in [1]. This study aims to find one or more DOP metrics that can be used to predict the navigation accuracy of any navigation system using carrier Doppler shift and pseudorange measurements. The study then considers how these DOP metrics can be used for LEO navigation constellation design. The OneWeb constellation is used as an example for constellation design with the DOP metrics.

Navigation systems using massive LEO constellations as an alternative to current Global Navigation Satellite Systems (GNSS)have been researched in a number of papers, including [2], [3], [4], [5], and [6]. In [2], the costs and benefits of using LEO satellites for navigation is discussed. The paper utilizes pseudorange measurements to analyze the potential of a navigation system based on LEO satellites. Using LEO satellites to augment GNSS is examined in [3]. In [4], a point-solution for position and velocity is proposed using only carrier Doppler shift from massive LEO constellations. Also, proposed in [4] is a GDOP analysis using only carrier Doppler shift. The combined pseudorange and carrier Doppler shift GDOP analysis in [1] is an attempted extension of the Doppler-only GDOP analysis in [4]. In [5] and [6], carrier Doppler shift measurements from LEO satellites are fused with measurements from an Inertial Navigation System (INS) to navigate with fewer than the 8 simultaneous measurements required for a carrier-Doppler-shift-point-solution. Carrier Doppler shift provides the capability to produce accurate position, velocity, and receiver clock offset rate estimates in [4], [5], and [6]. Unfortunately, the receiver clock estimate is orders of magnitude less accurate than traditional MEO-based GNSS when using only carrier Doppler shift. Driven by this fact, a proposal is made to include pseudorange measurement capabilities on the signals from a subset of the LEO satellites,and a GDOP analysis using both pseudorange and carrier Doppler shift measurements is developed in [1]. As shown in [1], one or two pseudorange measurements can significantly reduce the receiver clock offset error.

The present study considers DOP metrics for a combined pseudorange and carrier Doppler shift system and how they can be used for constellation design. DOP is a commonly used measure of accuracy for navigation systems. The traditional DOP analysis uses only pseudorange [7]. A GDOP analysis using only carrier Doppler shift is developed in [4]. A problem in using pseudorange and carrier Doppler shift is the need for unit agreement in a DOP analysis. Achievement of the needed agreement produces a non-dimensional measurement sensitivity matrix and a corresponding non-dimensionalized approximation of the resultant batch filter estimation error covariance matrix. A suitable non-dimensionalized error covariance matrix can be used to define a sensible DOP parameter. None of the methods for non-dimensionalization in [1] produce a single useful DOP metric for all scenarios. This study attempts to determine a single non-dimensionalized DOP parameter that can be used for all scenarios. In addition to the creation of a DOP analysis to quickly characterize the navigation accuracy of a given system, this study aims to perform basic constellation design using its DOP analysis to help identify driving error factors in a combined pseudorange and carrier Doppler shift LEO navigation system that has only a sparse set of pseudorange-capable satellites.

The development of a single DOP analysis that works for all scenarios is difficult due to the varying units and magnitudes of the values in the DOP matrix. A number of methods for non-dimensionalizing the terms in a GDOP analysis are developed in[1]. To find a method that works for all cases, this study splits the GDOP into two separate DOP metrics. One characterizes the accuracy of the position, velocity, and clock offset rate of a combined pseudorange and Doppler-shift navigation system. The other DOP metric characterizes only the receiver clock offset accuracy.

This study uses simulated data from LEO communication constellations such as Starlink and OneWeb. The simulated data are used by a batch filter to determine the position/clock-offset/velocity/clock-offset-rate navigation solution error and its corresponding error covariance matrix. Results from the batch filter are used to verify that the DOP analyses produce accurate error estimates. Due to the use of two DOP metrics, two separate maps of DOP vs. latitude and longitude are created for each constellation. While more data to consider than a single GDOP map, the two DOP maps provide concise information on the accuracy of navigation for a combined pseudorange and carrier Doppler shift LEO navigation system. The use of the two DOP metrics allows this study to employ a single DOP non-dimensionalization method for a combined pseudorange and carrier Doppler shift system.

This study also performs basic constellation design using the OneWeb constellation as an example. The study considers how many and which satellites need to include a pseudorange capability in order to reduce receiver clock offset error significantly over the entire Earth. Also, the study investigates the benefits of alternating versus grouping of ascending and descending nodes among other constellation design characteristics. The quick and simple DOP analyses allow for the rapid evaluation of changes in navigation accuracy due to variations of the simulated constellations.

REFERENCES

[1] B. McLemore and M. L. Psiaki, “GDOP of navigation using pseudorange and Doppler shift from a LEO constellation,” in Proceedings of the ION GNSS+ 2021, 9 20-24, 2021, pp. pp. 2783–2803.

[2] T. Reid, A. Neish, T. Walter, and P. Enge, “Broadband LEO constellations for navigation,”Navigation, vol. 65, no. 2, pp.205–220, Summer 2018.

[3] P. Iannucci and T. Humphreys, “Economical fused LEO GNSS,” in Proceedings of 2020 IEEE/ION Position, Location and Navigation Symposium (PLANS), 04 2020, pp. 426–443.

[4] M. L. Psiaki, “Navigation using carrier Doppler shift from a LEO constellation: TRANSIT on steroids,”NAVIGATION,vol. 68, no. 3, pp. 621–641, 2021. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/navi.438

[5] J. J. Morales, J. Khalife, A. A. Abdallah, C. T. Ardito, and Z. M. Kassas, “Inertial navigation system aiding with Orbcomm LEO satellite Doppler measurements,” in Proceedings of the 31st International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2018), Miami, Florida, 2018, pp. pp. 2718–2725.

[6] B. McLemore, “Navigation using Doppler shift from LEO constellations and ins data,” in Proceedings of the ION GNSS+2020, 9 22-25, 2020, pp. 3071–3086, virtual.

[7] P. Misra and P. Enge,Global Positioning System: Signals, Measurements, and Performance. Ganga-Jamuna Press, 2011.[Online]. Available: https://books.google.com/books?id=5WJOywAACAAJ

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