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### Session D6: Navigation Using Environmental Features

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**Measuring Gravitational Acceleration Using X-Ray Pulsars for Deep Space Navigation Algorithm Initialization**

*Kyle J. Houser and Demoz Gebre-Egziabher, University of Minnesota*

**Date/Time:** Friday, Sep. 20, 2:35 p.m.

I. BACKGROUND

Deep space navigation has been of increased interest with more missions targeting the Moon, Mars and beyond. With this renewed interest comes a drastic increase in the demand placed on navigation systems like the Deep Space Network (DSN). Proposed alternatives such as pulsar-based navigation (both in the x-ray and radio) could fill this gap but come with their own unique set of challenges (Mitchell et al., 2018; Runnels & Gebre-Egziabher, 2017; Sheikh et al., 2006). Many of these navigation methods rely on fairly accurate initial position estimates and then further refine the navigation solution via methods for signal tracking and filtering techniques. However, these systems could be prone to the so-called “lost in space” or “cold start” problem where the user does not have an adequate initial position solution to be able to initialize the navigation algorithms and is therefore unable to position itself. Methods to resolve this issue have been proposed using gravitational acceleration measurements (Houser & Gebre-Egziabher, n.d.), but previous methods have either lacked the accuracy required for use in these techniques (Houser et al., 2022) or still required a highly accurate initial state estimate to be able to initialize a digital phase lock loop (DPLL) (Anderson et al., 2022). In this paper, we aim to propose a method of measuring gravitational acceleration from x-ray pulsar time of arrival measurements that does not require an initial state estimate to work and is accurate enough to use in filtering techniques to provide an accurate initial position estimate to be able to initialize other navigation algorithms. We also present a Cramer-Rao Lower Bound analysis which aims to define the lower bound on acceleration measurement accuracy using this method. This method will also be validated using both Monte Carlo simulations as well as real x-ray photon observation data. By providing this acceleration measurement and fusing it with the filtering technique shown in Houser and Gebre-Egziabher (n.d.), various PNT algorithms could be cold started in deep space even in the face of loss of previous navigation state estimates.

II. METHODOLOGY

Tracking periodic signals is at the heart of radio-navigation systems. However, the x-ray pulsar signals of interest are incredibly weak which renders many standard signal tracking techniques such as those used in GNSS applications ineffective. To illustrate how weak these signals are, a representative observation of one full phase of the Crab Pulsar is shown in Figure 1. What an X-ray PNT sensor “sees” are discrete photons and not the signal profile (blue line in Figure 1) which encodes position information in its phase. There are multiple ways to deal with this challenge such as long duration time averaging (also known as epoch folding (Emadzadeh & Speyer, 2011)) or by analyzing the statistics of the photon arrivals to derive an estimator for various signal parameters Golshan and Sheikh (2007).

1. Pulsar Signal Parameters

We must first determine how estimates in pulsar signal parameters translate to measurements of position, velocity and acceler- ation. We can start by noting that a phase and Doppler shift of a pulsar signal encodes information about the kinematics of the observer. We use the relationship in Ashby and Golshan (2008) for the phase offset due to the observer position. We turn to the Doppler equation to show the relationship between frequency shift and velocity of the observer. Finally, because we desire an estimate of acceleration, we need to look at a frequency derivative of the pulsar signal. To determine the effect of the kinematic states of the observer on the various signal parameters, we can take a time derivative of the above equation. If we are able to jointly estimate phase, frequency and the first frequency derivative, we can form an estimate of position, velocity and acceleration by rearranging these equations.

2. Maximum Likelihood Estimator for Pulsar Signal Parameters.

We now turn to the method for estimating the the pulsar parameters of interest for our observer. To do this, we modify the MLE as shown in Golshan and Sheikh (2007) to include the first frequency derivative in addition to phase and frequency. To derive this MLE, we first note that photon arrival rates can be described by a Non-Homogeneous Poisson Process (NHPP). By following the process used in Golshan and Sheik (2007) we can derive a Log Likelihood Function which we can maximize to estimate our pulsar phase, frequency and frequency derivative. In order to aide in this parameter search, we need to determine a reasonable search space. We can define our search space by settingour midpoint to be the expected pulsar period for a static and inertial observer which can be obtained from pulsar timing models. From there we can also define maximum and minimum expected velocity which would correspond to the maximum and minimum Doppler shift or range rate we can expect. For example, we know our velocity cannot exceed the speed of light. More reasonably, we cannot expect it to exceed one tenth of a percent of the speed of light (again, we note the Parker Solar Probe as our current upper bound on spacecraft velocity). This velocity can be used as our upper and lower bounds as if our spacecraft was traveling at this velocity directly toward or away from the pulsar. Additionally, if we know the spacecraft is orbiting around a planet, we can add that planet’s orbital velocity l from its ephemeris data into our bounds. With the estimator derived, we now move to defining the Cramer Rao Lower Bound (CRLB) for this estimation method.

III. CRAMER-RAO LOWER BOUND ANALYSIS

To determine the theoretical performance of an estimator of acceleration using x-ray pulsar measurements, we turn to Cramer- Rao Lower Bound (CRLB) analysis (Kay, 1993). Ashby and Golshan (2008) presents a version of this lower bound analysis for the MLE that estimates position and velocity using photon time of arrival measurements from a pulsar signal, but we must redo this analysis since we have the additional parameter of the first frequency derivative. However, this provides a starting point to do this analysis. Additionally, the aspect of time has not been included in this analysis. This is an important aspect due to the fact that the accuracy of these measurements is strongly tied to the observer’s ability to accurately time the arrival of incoming photons. Therefore analyzing the effect of clock errors on the acceleration measurements will provide valuable insight as to the quality of clock that is required to be able to accurately measure phase, frequency and frequency derivatives. This analysis for position, velocity and acceleration has been done but does not yet include time as a stochastic parameter.

Finding the CRLB involves determining the Fischer information matrix, I(?), given the log likelihood function for the stochastic variables we are trying to estimate. We then invert the Fischer information matrix to find the theoretical lower bound on estimate accuracy for our parameters. For the sake of brevity in this abstract, we will skip over the details involved with deriving the CRLB. Instead we present the final diagonal terms for the CRLB that correspond to the bound on position (the [1,1] term), velocity (the [2,2] term) and acceleration (the [3,3] term) measurement variance, though these require the entire Fischer information matrix to be derived before they can be solved for.

IV. SIMULATION VALIDATION

To validate this method, a Monte Carlo simulation environment was used. This included generating simulated measurements of photon times of arrival for an observer in a simulated dynamic system with constant acceleration, then processing these simulated measurements to generate estimates of range, range rate and acceleration using the MLE. The accuracy of each of these estimates is then be compared to the CRLB to determine if the estimator converges to this bound. This is done for a wide range of observation times. For the purposes of this simulation, the signal parameters for Pulsar J1939+2134 when viewed using the NICER instrument were used. These results show that for observation times greater than 10,000 seconds the MLE converges to the CRLB and produces estimates of 1e-7 km/s2. These results are encouraging and motivate the need for experimental validation using real observation data of various pulsars.

V. EXPERIMENTAL VALIDATION

While simulation results show valuable insight into the validity of the estimator, it is still desired to be able to use this estimator on real data collected by observing pulsars on orbit. To do this, we will use this estimator on real data collected by the Neutron Star Interior Composition Explorer (NICER) observatory and the Chandra x-ray telescope. The Chandra telescope provides an interesting case for testing due to its highly eccentric orbit which allows for long continuous observation times for a single pulsar, but Runnels and Gebre-Egziabher (2021) notes that finding valid data sets is difficult due to a hardware error which degraded event time tagging. The NICER observatory, alternatively, offers incredible timing accuracy but lacks long duration observations due to its orbit on the International Space Station in low-earth orbit (Prigozhin et al., 2016). Through the post processing of both of these datasets, we hope to be able to accurately determine the efficacy of this estimator using real data.

REFERENCES

Anderson, K. D., Pines, D. J., & Sheikh, S. I. (2022). Investigation of X-ray Pulsar Signal Phase Tracking for Spacecraft Navigation. AIAA Science and Technology Forum and Exposition, AIAA SciTech Forum 2022. https://doi.org/10.2514/6.2022-1589

Ashby, N., & Golshan, A. R. (2008). Minimum uncertainties in position and velocity determination using x-ray photons from millisecond pulsars. Proceedings of the Institute of Navigation, National Technical Meeting, 1, 110–118.

Emadzadeh, A. A., & Speyer, J. L. (2011). X-Ray Pulsar-Based Relative Navigation using Epoch Folding. IEEE Transactions on Aerospace and Electronic Systems, 47(4), 2317–2328. https://doi.org/10.1109/TAES.2011.6034635

Fishbane, P., Gasiorowicz, S., & Thornton, S. (1993). Physics for Scientists and Engineers (2nd ed., Vol. 2). Prentice Hall. Golshan, A. R., & Sheikh, S. I. (2007). On pulse phase estimation and tracking of variable celestial X-ray sources. Proceedings of the Annual Meeting - Institute of Navigation.

Houser, K. J., & Gebre-Egziabher, D. (n.d.). Algorithm for Initializing Deep Space PNT Methods Using Gravitational Acceleration Measurements. NAVIGATION (In Review).

Houser, K. J., Runnels, J. T., & Gebre-Egziabher, D. (2022). Gravitational Acceleration Maps for Initialization and Ambiguity

Resolution of Pulsar-Based PNT in Space. 35th International Technical Meeting of the Satellite Division of the Institute of Navigation, ION GNSS+ 2022, 3. https://doi.org/10.33012/2022.18542

Kay, S. M. (1993). Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Pearson.

Lyne, A. G., Pritchard, R. S., & Smith, F. G. (1993). 23 years of Crab pulsar rotational history. Monthly Notices of the Royal Astronomical Society, 265. https://doi.org/10.1093/mnras/265.4.1003

Mitchell, J. W., Winternitz, L. B., Hassouneh, M. A., Price, S. R., Semper, S. R., Yu, W. H., Ray, P. S., Wolff, M. T., Kerr, M., Wood, K. S., Arzoumanian, Z., Gendreau, K. C., Guillemot, L., Cognard, I., & Demorest, P. (2018). Sextant X-Ray pulsar navigation demonstration: Initial on-orbit results. Advances in the Astronautical Sciences, 164.

Prigozhin, G., Gendreau, K., Doty, J. P., Foster, R., Remillard, R., Malonis, A., LaMarr, B., Vezie, M., Egan, M., Villasenor, J., Arzoumanian, Z., Baumgartner, W., Scholze, F., Laubis, C., Krumrey, M., & Huber, A. (2016). NICER instrument detector subsystem: description and performance. Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, 9905. https://doi.org/10.1117/12.2231718

Runnels, J. T., & Gebre-Egziabher, D. (2017). Recursive range estimation using astrophysical signals of opportunity. Journal of Guidance, Control, and Dynamics, 40. https://doi.org/10.2514/1.G002650

Runnels, J. T., & Gebre-Egziabher, D. (2021). Estimator for Deep-Space Position and Attitude Using X-ray Pulsars. IEEE Transactions on Aerospace and Electronic Systems, 57. https://doi.org/10.1109/TAES.2021.3068432

Sheikh, S. I., Pines, D. J., Ray, P. S., Wood, K. S., Lovellette, M. N., & Wolff, M. T. (2006). Spacecraft navigation using x-ray pulsars. https://doi.org/10.2514/1.13331

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