Performance Analysis of Simultaneous Tracking and Navigation with LEO Satellites

Trier R. Mortlock and Zaher M. Kassas University of California, Irvine, USA

BIOGRAPHIES Trier R. Mortlock is a Ph.D. student in the Department of Mechanical and Aerospace Engineering at the University of California, Irvine and a member of the Autonomous Systems Perception, Intelligence, and Navigation (ASPIN) Laboratory. He received a B.S. in Mechanical Engineering from the University of California, Berkeley. He serves in the U.S. Army Reserve serving as a Cyber Operations Officer. His current research interests include cyber-physical systems, satellite-based navigation, and situational awareness in dynamic uncertain environments.

Zaher (Zak) M. Kassas is an associate professor at the University of California, Irvine and director of the ASPIN Laboratory. He received a B.E. in Electrical Engineering from the Lebanese American University, an M.S. in Electrical and Computer Engineering from The Ohio State University, and an M.S.E. in Aerospace Engineering and a Ph.D. in Electrical and Computer Engineering from The University of Texas at Austin. In 2018, he received the National Science Foundation (NSF) Faculty Early Career Development Program (CAREER) award, and in 2019, he received the Office of Naval Research (ONR) Young Investigator Program (YIP) award. He is a recipient of 2018 IEEE Walter Fried Award, 2018 Institute of Navigation (ION) Samuel Burka Award, and 2019 ION Col. Thomas Thurlow Award. He is an Associate Editor for the IEEE Transactions on Aerospace and Electronic Systems and the IEEE Transactions on Intelligent Transportation Systems. His research interests include cyber-physical systems, estimation theory, navigation systems, autonomous vehicles, and intelligent transportation systems.

ABSTRACT An opportunistic navigation framework using low Earth orbit (LEO) satellites is analyzed. This framework, termed simultaneous tracking and navigation (STAN), estimates a navigating vehicle’s state along with the states of orbiting LEO satellites. STAN employs an extended Kalman filter (EKF) to fuse measurements from satellite receivers and an inertial navigation system (INS). The navigation performance is analyzed due to: (i) the vehicle being equipped with (1) different inertial measurement unit (IMU) grades: consumer, industrial, and tactical and (2) different receiver clock quality: temperature-compensated crystal oscillators (TCXO) and oven-controlled crystal oscillators (OCXO) and (ii) the LEO satellites being equipped with different transmitter clock quality: OCXO and chip-scale atomic clock (CSAC). Additionally, the effect of utilizing a large number of LEO satellites for navigation is investigated. This analysis provides insight into the achievable performance of STAN, which can serve as an alternative navigation system in global navigation satellite system (GNSS)-denied environments. The performance predictions from simulations are compared with experimental results with real signals from the Orbcomm LEO constellation. A close match between the simulation and experimental results is demonstrated for an unmanned aerial vehicle (UAV) navigating via the STAN framework with signals from two Orbcomm LEO satellites for 160 seconds, the last 35 seconds of which are without GNSS signals. The UAV’s position root-mean squared error (RMSE) from simulations was 8.6 m, while the experimental position RMSE was 10 m.

I. INTRODUCTION

There has been a surge in recent years to establish resilient positioning, navigation, and timing (PNT) services which possess features of accessibility and integrity [1]. This surge embodies the paramount need for resilient PNT on numerous critical infrastructure (e.g. transportation systems, power grids, communications, military operations, emergency response missions) that rely on global navigation satellite systems (GNSS), which are vulnerable to inter- ference, jamming, and spoofing [2]. This paper examines the use of low Earth orbit (LEO) satellites for navigation purposes in GNSS-denied environments.

Copyright c 2020 by T. R. Mortlock Preprint of the 2020 ION GNSS+ Conference and Z. M. Kassas St. Louis, Missouri, September 21–25, 2020 The paper focuses in particular on unmanned aerial vehicles (UAVs), which traditionally navigate their trajectories by relying on a tightly coupled system of inertial measurement units (IMUs), used for short-term positioning updates and a local navigation solution, and GNSS signals, used to correct accumulated errors from IMU measurements and to provide a navigation solution in a global frame. This traditional framework faces challenges as GNSS signals de- grade when navigating indoors, in deep urban canyons, or under dense foliage, reducing the accuracy and availability of the navigation system. Degradation poses serious safety risks for many navigation missions, including aviation, transportation, disaster relief, and military operations. Furthermore, GNSS signals are prone to unintentional in- terference, intentional jamming, or malicious spoofing, which could have catastrophic consequences [3]. The idea of exploiting ambient radio frequency signals of opportunity for navigation has been an area of extensive recent study [4–6]. Past works have exploited terrestrial signals from AM/FM radio, cellular, and digital television signals [7–14], as well as non-terrestrial signals from satellite constellations [15–21] for navigation purposes. Meter-level accurate navigation has been demonstrated on ground vehicles with terrestrial cellular and television signals [22–24], while sub-meter-level accurate navigation has been demonstrated on UAVs [25, 26]. Moreover, opportunistic navigation with LEO signals has been demonstrated on ground vehicles and UAVs with existing constellations, while showing the potential of achieving sub-meter level accuracy with future megaconstellation LEO satellites [27].

LEO satellites offer a promising source of signals to leverage opportunistically for navigation purposes. There are currently over 1,900 LEO satellites in operational orbits, and numerous companies like SpaceX, Samsung, Boeing, and OneWeb are engaged in launching tens of thousands more over the next decade [28]. Fig. 1 shows a subset of 717 active LEO satellites from the following six constellations: Starlink, OneWeb, Iridium, GlobalStar, Iridium Next, and Orbcomm [29]. The surge to add to the current LEO satellites is evident in the recent request made on behalf of SpaceX to add 30,000 LEO satellites to their current efforts [30]. LEO satellite signals offer a number of unique benefits for navigation purposes: (i) strong signal strength due to their lower orbits, (ii) diversity in their geometries and frequencies, (iii) availability from different orbits and constellations, and (iv) the ability to observe and collect free of charge, with the proper receivers. However, the use of LEO satellites for navigation comes with several challenges, most notably: (i) the satellites cannot be assumed to be transmitting their states, (ii) the satellites cannot be assumed to be equipped with atomic oscillators, nor to be tightly synchronized, and (iii) extracting navigation observables from LEO satellites is not yet fully understood. This paper focuses on the first two challenges by adopting the simultaneous tracking and navigation (STAN) framework. The STAN framework estimates the dynamic, stochastic states of the LEO satellites simultaneously with the states of the navigating vehicle [20]. STAN utilizes a filter (e.g. extended Kalman filter) that couples GNSS and LEO receivers with an IMU. STAN considered a simplified LEO satellite dynamical model in [20] and was adapted to account for the case where LEO satellites periodically transmit their positions in [31]. More elaborate LEO satellite dynamics models were studied in [32]. A differential framework was proposed in [21, 33].

Fig. 1. Subset of 717 active LEO satellites from the following six constellations: Starlink, OneWeb, Iridium, GlobalStar, Iridium Next, and Orbcomm [29] from June 2020. Map data: Google Earth.

This paper presents an analysis of a vehicle’s navigation performance while operating via the STAN framework using LEO satellite signals. This type of performance characterization is vital to understanding the viability of this navigation framework and feasibility of its use under various conditions. The effect of the quality of the navigating vehicle’s IMU grade and both the quality of the vehicle-mounted oscillator and LEO satellites’ oscillators on the navigation solution is studied. Bounds for STAN’s performance as a function of the sensor types used, as well as the number of LEO satellites used, are presented through simulation. Experimental results that utilize the STAN framework for navigation is compared with the paper’s proposed performance characterization, showing a close match between the simulation and experimental results. These results demonstrate a UAV navigating via the STAN framework with signals from two Orbcomm LEO satellites for 160 seconds, the last 35 seconds of which are without GNSS signals. The UAV’s position root-mean squared error (RMSE) from simulations was 8.6 meters, while the experimental position RMSE was 10 meters. The future of high-availability navigation requires overcoming both purposeful and incidental GNSS degradation alike, and the analysis of navigation with LEO satellites under the STAN framework helps assess the feasibility of future alternative navigation sources.

The remainder of this paper is organized as follows. Section II details the STAN framework. Section III characterizes the effect of sensor errors on STAN’s navigation performance through simulations. Section IV compares the simula- tor’s performance to experiments conducted on a UAV navigating with real LEO signals via the STAN framework. Section V contains concluding remarks.

II. STAN FRAMEWORK

The STAN framework, depicted in Fig. 2, utilizes an extended Kalman filter (EKF) that tightly couples measurements made from an IMU, a GNSS receiver, and a LEO satellite receiver. A key difference between STAN and a traditional tightly coupled GNSS-INS system is the estimation of the LEO satellites’ positions, velocities, and clocks along with the vehicle’s states. During the prediction stage of the EKF; IMU, clock, and LEO propagation models are used to predict the filter’s state. Subsequently, during the update stage of the EKF, the filter uses measurements from the LEO receiver and the GNSS receiver to update the state estimates. When GNSS satellites are no longer available, the filter continues to estimate the states of both the vehicle and the LEO satellites, using only updates from the LEO satellites. The filter states and the corresponding dynamics and measurement models of the filter are discussed next. Following this, an overview of LEO satellite megaconstellations and simulations using STAN are presented.

EKF Prediction GNSS IMU INS Receiver

State LEO EKF Propagation Initialization Update

Orbit Clock Models LEO Receiver Determination

Fig. 2. STAN framework that uses LEO satellite signals and GNSS signals (when available) to simultaneously tracks that states of LEO satellites while aiding a vehicle’s INS [20].

A. STAN Models The state vector of the EKF is defined as follows: T T T T x = xr , xleo,1,..., xleo,M (1) T  B T T T T T  ˙ xr = Gq¯ , rr , r˙r , bg , ba , cδtr, cδtr (2) h iT T T ˙ xleo,m = rleo,m, r˙leo,m, cδtleo,m, cδtleo,m (3) h i B where xr is the state vector of the vehicle consisting of Gq¯, a four-dimensional (4-D) unit quaternion vector rep- resenting the orientation of a body frame B fixed at the IMU with respect to a global frame G; rr and r˙r, the three-dimensional (3-D) position and velocity vector of the vehicle, respectively; bg and ba, the 3-D biases of the IMU’s gyroscope and accelerometer, respectively; δtr and δt˙ r, the clock bias and drift of the receiver, respectively; th and c is the speed of light. The vector xleo,m is the state vector of the m LEO satellite, consisting of rleo,m and r˙leo,m, the 3-D satellite position and velocity, respectively; δtleo,m and δt˙ leo,m, the satellite’s transceiver clock bias and drift, respectively; and m =1,...,M, with M being the total number of LEO satellites used under STAN.

The EKF propagates the vehicle’s position, velocity, and orientation using measurements from the IMU processed with the strap-down INS kinematic equations [34]. The vehicle’s accelerometer and gyroscope biases, covered in Section III, are propagated according to a velocity random walk model. The clock states of the vehicle and LEO satellites, also covered in Section III, are propagated through a double integrator model with a specified process noise. The LEO satellite position and velocity are predicted through a two-body with J2 propagation model, where J2 is the second gravitational zonal coefficient [32]. During the EKF update, the vehicle-mounted LEO satellite receiver makes pseudorange and pseudorange rate measurements from each satellite. The Doppler frequency measurements fD are made from the transmitted LEO satellite signals, from which a pseudorange rate measurementρ ˙ can be obtained asρ ˙ = − c f , where f is the carrier frequency. The pseudorange measurement ρ at time-step j from fc D c leo,m the mth LEO satellite is modeled according to:

ρleo,m(j)= krr(j) − rleo,m(j)k2 + c · [δtr(j) − δtleo,m(j)] + vρleo,m (j), j =1, 2,..., (4) where the common ionospheric and tropospheric delay components are not included due to their negligible effects compared to the LEO satellite position and velocity estimate errors [35], and vρleo,m is the measurement noise, which 2 is modeled as a white Gaussian random sequence with variance σρleo,m . The LEO receiver also makes pseudorange rate measurements,ρ ˙leo,m, on the LEO satellites which are modeled following the same above assumptions as

T [rr(j) − rleo,m(j)] ˙ ˙ ρ˙leo,m(j) = [r˙leo,m(j) − r˙r(j)] + c · [δtr(j) − δtleo,m(j)] + vρ˙leo,m (j), j =1, 2,..., (5) krr(j) − rleo,m(j)k2

2 where vρ˙leo,m is the measurement noise, which is modeled as a white Gaussian random sequence with variance σρ˙leo,m . Ultimately, the EKF outputs an estimate of the state vector, denoted xˆ(k), and a corresponding estimation error covariance matrix, denoted P(k). The estimation error covariance matrix is an important representation of the achievable navigation performance of a system and provides bounds for the filter’s estimation error.

B. LEO Satellite Megaconstellations There are a variety of companies operating satellites in LEO space, with numerous new constellations vying for their respective share of orbits. Established and traditionally communications-based constellations (e.g., Orbcomm, Glob- alstar, Iridium) are being joined by new waves of constellations (e.g., SpaceX’s Starlink, Amazon’s Project Kuiper, SpaceMobile, Telesat) aiming to provide broadband internet to the world through megaconstellations of thousands of LEO satellites. The specific constellation used in this section is the Starlink constellation of LEO satellites, operated by SpaceX, which has already established a significant presence in LEO. The North American Aerospace Defense Command (NORAD) maintains a publicly available database of current orbiting satellites composed of two-line ele- ment (TLE) files which contain ephemeris data for each satellite [29]. For this section, the ground truth of the satellite positions was generated by pulling data from the NORAD web-server and utilizing a two-body with J2 perturbations orbit propagation model to generate the satellite trajectories. This orbit model is a first-order approximation that builds upon the traditional two-body model with the added J2 component. The general two-body model is written as [32] dUm r¨leo,m = agravm , agravm = , (6) drleo,m with Um representing the non-uniform gravity potential of the Earth . In [36], the JGM-3 model for Um is presented, and after dropping lower magnitude terms such as the tesseral and sectoral factors, Um at each LEO satellite can be written as [37] N µ Rn U = 1 − J E P (sin(θ)) , (7) m kr k n kr kn n leo,m " n=2 leo,m # X th where Pn is a Legendre polynomial with harmonic n, Jn is the n zonal coefficient, RE is the mean radius of the T th Earth, sin(θ)= zleo,m/krleo,mk, rleo,m , [xleo,m, yleo,m, zleo,m] are the position coordinates of the m LEO satellite in an Earth-centered inertial frame, and N = ∞.

This section uses data from late May 2020, when there were over 400 Starlink satellites in operational orbits. In June 2020, SpaceX conducted more launches pushing their total to over 500 LEO satellites. Fig. 3(a) shows simulated orbits of 118 of these active satellites that were visible over an elevation mask of 5◦ degrees for a stationary receiver at the University of California, Irvine (UCI) at some point during the satellites orbital period. The red sections of the orbits indicated when the satellites were visible over the elevation mask. Fig. 3(b) shows a heat map of the future Starlink megaconstellation that will be visible above a 5◦ elevation mask at a given point in time over Earth [21]. Although some current LEO constellations like Orbcomm transmit signals which are available at low elevation angles, collection of signals from constellations like Starlink has not been studied, to the authors’ knowledge. In this study, signal availability for exploitation is assumed to be possible at these lower elevation angles.

(a) (b) Fig. 3. (a) Simulation of 118 current active Starlinks over UC Irvine for an elevation mask of 5◦. (b) Heat map of the future number of visible Starlink LEO satellites at any point on Earth for an elevation mask of 5◦. (With permission from [21].)

C. Simulation Overview The subsection gives an overview of the base scenario used for the simulations in this paper. A fixed-wing UAV, equipped with an IMU as well as GNSS and LEO receivers, navigates with signals from a varying number of LEO satellites from the Starlink megaconstellation. The LEO receiver was assumed to produce pseudorange and Doppler measurements to visible Starlink LEO satellites. The pseudorange and pseudorange rate measurement noise variances ranged between 1.67–4.64 m2 and 0.37–0.92 (m/s)2, respectively, which were varied based on the predicted carrier-to-noise ratio (C/N0), as calculated based on the satellites’ elevation angle. The simulated UAV compares in performance to a small private plane with a cruise speed of roughly 50 m/s. The UAV flies a 360-second trajectory covering 21.8 km, shown in white in Fig. 4(a), consisting of a straight climbing segment, followed by a figure-eight pattern over Irvine, California, USA, and then a final descent into a straight segment. The UAV, initially at 1 km, climbs to an altitude of 1.5 km, where it begins executing rolling and yawing maneuvers before descending back down to 1 km in the straight segment. The LEO satellite states are initialized using TLE files and the trajectories of the 20 Starlink LEO satellites used for navigation are shown in white in Fig. 4(b). GNSS was available for the first 60 seconds of the flight, and STAN’s estimate for the satellites are displayed in green during this 60-second time period. The final 300 seconds of tracking and navigation without GNSS are show in red in Fig. 4(a,b). Fig. 4(c,d) also shows a zoom of the final trajectory estimate of the vehicle and one of the LEO satellites.

III. SENSOR ERROR CHARACTERIZATION

Different navigating vehicles can be equipped with varying sensor suites that can have a significant effect on the vehicle’s positioning accuracy [38]. For example, tactical, industrial, and consumer are all different grades of IMUs (b)

(a)

(d)

(c)

Fig. 4. (a) UAV trajectory over Irvine, California, USA: truth (white) and STAN estimate (red). (b) 20 Starlink LEO satellite trajectories: truth (white), tracked during GNSS availability (green) and estimated without GNSS (red). (c) Zoom of final true UAV trajectory (white) and STAN estimate (red). (d) Zoom of final true LEO satellite trajectory (white) and STAN estimate (red). Map data: Google Earth. which contain different inherent noise statistics and parameters that affect the vehicle’s navigation solution. Similarly, temperature-compensated crystal oscillators (TCXO), oven-controlled crystal oscillators (OCXO), and chip scale atomic clocks (CSAC) are three different types of clocks that have differing timing stability parameters, which ultimately leads to different navigation performance. While investigating the performance of the STAN framework, it is important to study how different sensor parameter errors contribute to the PNT error of the navigating vehicle. This section establishes important performance characterizations that lead to mission-driven navigation requirements, while a vehicle navigates under STAN. The analysis in this section is performed for an illustrative scenario with a specific UAV trajectory. The sensors studied in this paper and the simulation settings are discussed next.

A. IMU Sensor Errors In a traditional strap-down inertial measurement navigation algorithm composed of a triad-gyroscope and a triad- accelerometer, the two sensors make measurements that are processed to produce a position estimate of the vehicle. The gyroscope measures the angular velocity of the vehicle in the body frame with respect to an inertial frame, and these measurements are then integrated to track the orientation of the vehicle. The accelerometer measures the specific force acting on the vehicle, which in turn is orientated to the inertial frame, corrected for gravity, and integrated twice to obtain the position estimate of the vehicle. There are various sensor errors that can distort accelerometer and gyroscope measurements, namely, biases, random noises, and scale factor errors [39]. Propagation of these errors through the integrations described above has compounding effects on the degradation of the IMU’s positioning estimate [40]. This paper focuses on bias errors and the measurement noise statistics of the vehicle’s IMU while navigating with LEO satellites under STAN. The triad-gyroscope and triad-accelerometer produces angular rate ωimu and specific force aimu measurements, modeled as B ωimu(k)= ω(k)+ bg(k)+ ng(k), k =1, 2,... (8)

Bk G G aimu(k)= R G q¯(k) a(k) − g(k) + ba(k)+ na(k), k =1, 2,..., (9)

Bk h i
G where R[G q¯] is the rotation matrix representation of the quaternion vector from the global to the body frame, g G and a are the acceleration and the gravity acceleration in the global frame, and ng and na are measurement noise 2 2 vectors, which are modeled as white noise sequences with covariances σg I3×3 and σa I3×3, respectively.

The gyroscope’s and accelerometer’s biases, bg and ba, respectively, are modeled to evolve according to a velocity random walk model

bg(k +1)= bg(k)+ wbg(k), k =1, 2,..., (10)

ba(k +1)= ba(k)+ wba(k), k =1, 2,..., (11) where wbg and wba are process noise vectors, which are modeled as a discrete-time white noise sequences with 2 2 covariances σbgI3×3 and σbaI3×3, respectively.

The following simulations study how different IMU biases and noise statistics affect the performance of a vehicle using STAN for navigation during GNSS unavailability. The UAV’s trajectory in these simulations is fixed, which is the trajectory detailed in Section II. The maneuvers a vehicle undergoes alter how the IMU errors propagate into the navigation solution, and for this reason, the vehicle’s motion along the chosen trajectory excites all three directions of both the accelerometer and gyroscope. The LEO satellites orbits are the same as those outlined in ◦ Section II, and are propagated using a two-body with J2 model. An elevation mask of 16.5 was set to maximize LEO satellite availability for the cases of 0 to 50 LEO satellites. For the case of 100 LEO satellites, due to a lack of total visible satellites, an elevation angle of 10◦ was used with a combination of the current Starlink satellites and future projected Starlink satellites based on proposals made to the Federal Communications Commission [41–43]. The number of chosen LEO satellites was determined in order to illustrate the navigation advantages of increasing the number of LEO satellites used under the STAN framework. The UAV and LEO satellites were equipped with a high-quality TCXO and high-quality OCXOs, respectively, and GNSS was cut after 60 seconds. The IMU grade the vehicle was equipped with was varied from highest to lowest performance: tactical, industrial, and consumer. During each simulation the biases and noise parameters for each IMU grade were altered to those listed in Table I. Note that for the vectors bg and ba, the values listed in the tables are the same for each of the variable’s three dimensions. Fig. 5 illustrates the ±3σ estimation error bounds of the EKF for the UAV’s states in the East-North-Up local coordinate frame, and the estimation error trajectories are not plotted in these figures in order not to convolute the plots. Each color represents a different number of LEO satellites used for tracking and navigation, and appears on the plots three times for each of the different IMU grades. The dashed vertical line at 60 seconds on each plot represents the GNSS cutoff time. The relatively sharp decrease around 160 seconds, most easily observed in the case of 5 LEO satellites with a consumer grade IMU in the north direction of Fig. 5, is due to the UAV’s trajectory. The wave-like increasing and decreasing values observed in the orientation states of the vehicle in Fig. 5 are attributed to the banking turns in the vehicle’s trajectory. The following observations consider the first three plots, representing the uncertainty in the vehicle position states. As expected, after GNSS is cut off without any LEO satellites, the uncertainty diverges very quickly. As five LEO satellites are used, the uncertainty diverges considerably slower compared to no satellites at all, since the LEO measurements reduce the divergence caused by IMU error propagations. When 20 LEO satellites are used, an industrial-grade IMU provides comparable performance to a tactical-grade IMU. Additionally, when 50 LEO satellites are used, in the North position error, a consumer-grade IMU performs similarly to a tactical-grade IMU with 20 LEO satellites. Furthermore, when 100 LEO satellites are used, a consumer-grade IMU provides roughly the same performance as both the industrial-grade and tactical-grade IMUs. These results help determine achievable navigation performance using STAN as a function of time without GNSS with different numbers of LEO satellites and IMU grades.

TABLE I IMU Grade Simulation Settings Parameters Units Tactical Industrial Consumer −4 −4 −3 σg rad/s 2.036 × 10 2.891 × 10 5.236 × 10 −6 −4 −3 bg(0) rad/s 4.848 × 10 2.417 × 10 2.9 × 10 −7 −7 −5 σbg rad/s 1 × 10 5 × 10 1 × 10 2 −6 −6 −2 σa m/s 1.629 × 10 4.062 × 10 2.452 × 10 2 −3 −2 −1 ba(0) m/s 1.962 × 10 2.943 × 10 7.848 × 10 2 −7 −7 −5 σba m/s 1 × 10 5 × 10 1 × 10 Number of LEO Satellites / IMU Grade 0 / Tactical 0 / Industrial 0 / Consumer 5 / Tactical 5 / Industrial 5 / Consumer 20 / Tactical 20 / Industrial 20 / Consumer 100 / Tactical 100 / Industrial 100 / Consumer 500

250 GPS cutoff 0

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10

0

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10

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0 Yaw [rad] -0.5 0 50 100 150 200 250 300 350 Time [s] Fig. 5. EKF ±3σ estimation error bounds of the UAV’s states in the local navigation frame for varying IMU grades and number of LEO satellites. The UAV is assumed to be equipped with a high-quality TCXO, while LEO satellites were equipped with high-quality OCXOs. B. Clock Errors Clock performance in navigation systems is typically characterized by the type of oscillator used such as TCXO, OCXO, or CSAC. Clock error modelling and characterization has been studied extensively for GNSS, and recent advances in oscillators have made higher quality clocks more affordable and attainable in terms of size and power [44, 45]. Oscillators produce sine waves which deviate from their nominal frequencies due to corruption from noise. The Allan variance is a metric that measures the stability of an oscillator by comparing the measured frequency to that of the nominal, whereafter the power spectrum is determined. The noise power spectrum has been shown through laboratory experiments to be a combination of power-law coefficients involving phase and frequency noises [46]. Under STAN, the same clock model for both the vehicle and the LEO satellites is assumed to evolve according to xclk (k +1)= Fclk xclk(k)+ wclk(k), k =1, 2,..., (12) T 1 T x , cδt, cδt˙ , F = , clk clk 0 1 h i   where T is the constant sampling interval and wclk is the process noise [47], which is modeled as a discrete-time white noise sequence with covariance

T 3 T 2 Sw˜δt T + Sw˜δ˙t 3 Sw˜δ˙t 2 Qclk = T 2 . (13) " Sw˜δt˙ 2 Sw˜δt˙ T #

The terms Sw˜δt and Sw˜δt˙ are the clock bias and drift process noise power spectra, respectively, which can be related 2 to the power-law coefficients, {hα,}α=0,−2, which characterizes the power spectral density of the fractional frequency h0 2 ≈ ≈ − deviation of an oscillator from nominal frequency according to Sw˜δt 2 and Sw˜δt˙ 2π h 2 [46]. In the following simulations, the UAV and LEO satellites followed the same trajectories as detailed in Section II. The UAV was equipped with a tactical-grade IMU. The navigation performance was studied by varying the used oscillators, while also increasing the number of LEO satellites. Table II details the three different types of oscillators used in this section [48]. Fig. 6 shows the ±3σ estimation error bounds of the UAV while setting the LEO clocks as high-quality OCXOs, and varying the number of satellites and varying the clock used by the UAV among high- quality TCXO, typical-quality OCXO, and high-quality OCXO. The different colors correspond to different numbers of LEO satellites, and each color is represented on the plots three times with a different line style for the varying UAV oscillator quality. It is evident from Fig. 6 that the performance with the OCXOs are similar, while the TCXO diverges noticeably faster. With 100 LEO satellites, the performance with a TCXO becomes comparable to the OCXOs. Fig. 7 shows the effect of varying the LEO satellite clocks as CSACs, high-quality OCXOs, and typical-quality OCXO while the UAV oscillator is held constant as a high-quality TCXO. An interesting finding is that regardless of the number of LEO satellites used, the vehicles navigation performance is surprisingly not sensitive to the LEO clock quality.

TABLE II Clock Quality Simulation Settings

Quality parameters {h0, h−2} High-quality TCXO 9.4 × 10−20, 3.8 × 10−21 −20 −23 Typical-quality OCXO 8.0 × 10 , 4.0 × 10 −22 −26 High-quality OCXO 2.6 × 10 , 4.0 × ·10 −21 −27 CSAC  7.2 × 10 , 2.7 × 10  Number of LEO Satellites / UAV Clock Type 0 / High-quality OCXO 0 / Typical-quality OCXO 0 / High-quality TCXO 5 / High-quality OCXO 5 / Typical-quality OCXO 5 / High-quality TCXO 20 / High-quality OCXO 20 / Typical-quality OCXO 20 / High-quality TCXO 100 / High-quality OCXO 100 / Typical-quality OCXO 100 / High-quality TCXO 500 GPS cutoff 250

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50 0 -50

-100 Clock Bias [m] 2

1

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Clock Drift [m/s] -2 0 50 100 150 200 250 300 350 0.5 Time [s] Fig. 6. EKF ±3σ estimation error bounds of UAV’s states in the local navigation frame for varying number of LEO satellites and UAV oscillator. The LEO satellite oscillators were fixed as high-quality OCXOs. The UAV was equipped with a tactical-grade IMU. Number of LEO Satellites / LEO Satellite Clock Type 0 / CSAC 0 / High-quality OCXO 0 / Typical-quality OCXO 5 / CSAC 5 / High-quality OCXO 5 / Typical-quality OCXO 20 / CSAC 20 / High-quality OCXO 20 / Typical-quality OCXO 100 / CSAC 100 / High-quality OCXO 100 / Typical-quality OCXO 500 GPS cutoff 250

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-2.5

Clock Drift [m/s] -5 0 50 100 150 200 250 300 350 0.5 Time [s] Fig. 7. EKF ±3σ estimation error bounds of UAV’s states in the local navigation frame for varying number of LEO satellites and LEO satellite oscillator. The UAV oscillator was fixed as a high-quality TCXO. The UAV was equipped with a tactical-grade IMU. IV. EXPERIMENTAL RESULTS

This section compares the simulation performance with experimental results, which used two Orbcomm LEO satel- lites. In the experiment, a DJI Matrice 600 UAV flew for 160 seconds via the STAN framework. GNSS signals were artificially cut off after 125 seconds. The UAV was equipped with a consumer-grade IMU, a pressure altimeter, a high-end quadrifilar helix antenna, and an Ettus E312 universal software radio peripheral (USRP). The Ettus E312 was used to sample Orbcomm signals and store the in-phase and quadrature components. These samples were then processed by a Multichannel Adaptive TRansceiver Information eXtractor (MATRIX) software-defined receiver (SDR) [20] to perform carrier synchronization and extract pseudorange rate observables. Finally, a Matlab-based estimator was used to implement the STAN algorithms for the UAV. Fig. 8 shows the experimental environment and navigation results. To assess the fidelity of the simulation results presented in Section III, the experimental results are compared with the simulate used throughout the paper, which was configured to use: two LEO satellites, tactical- grade IMU, UAV clock as high-quality TCXO, LEO satellite clocks as high-quality TCXOs. Table III compares the navigation results obtained from the simulator versus the experiment. To the authors’ knowledge, these are the first results demonstrating navigation of a UAV with real LEO satellite signals, along with side-by-side comparison with high-fidelity simulations.

Irvine, California Orbcomm LEO satellite 1 trajectory

Orbcomm LEO satellite 2 (a) trajectory (c)

(b)

Experiment Flight end RMSE: 10 m Final error: 20 m Flight start Simulation RMSE: 8.6 m GPS cutoff Final error: 18 m Truth (d)

Fig. 8. Experimental results showing (a) the trajectory of the 2 Orbcomm LEO satellites, (b) zoom on the UAV’s final position and final position estimates, and (c)–(d) true and estimated trajectories of the UAV undergoing a 160 second trajectory with GNSS cut for the last 35 seconds [32].

TABLE III UAV Navigation Performance

Performance Measure Simulation Experiment RMSE (m) 8.6 10 Final Error (m) 18 20

V. CONCLUSION

This paper analyzed the achievable navigation performance of STAN with LEO satellites signals. Simulations of active Starlink satellites were used to navigate a fixed-wing UAV over a trajectory covering 21.8 km over 360 seconds, the last 300 seconds of which was without GNSS signals. Different IMU grades were examined as a function of the gyroscope and accelerometer used. It was demonstrated that increasing the number of LEO satellites used compensates for using lower quality IMUs. Specifically, with 100 LEO satellites, a consumer-grade IMU performs comparably with a tactical-grade IMU. The effect of the vehicle’s clock and the LEO satellite clocks was studied, showing that both the number of LEO satellites and the quality of UAV-equipped clocks have a noticeable effect on navigation performance, while the LEO-equipped clocks had little influence on the UAV’s navigation performance. Experimental results were presented for a UAV navigating via the STAN framework with signals from two Orbcomm LEO satellites for 160 seconds, the last 35 seconds of which are without GNSS signals. The experimental results showed a close match with simulation results. The UAV’s position root-mean squared error (RMSE) from simulations was 8.6 m, while the experimental position RMSE was 10 m.

ACKNOWLEDGMENTS This work was supported in part by the Office of Naval Research (ONR) under Grant N00014-19-1-2511 and in part by the National Science Foundation (NSF) under Grant 1929965. The authors would like to thank Joshua Morales and Joe Khalife for helpful discussions.

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