Performance Analysis of Simultaneous Trackingand Navigation with LEO Satellites

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

BIOGRAPHIES

Trier R. Mortlock is a Ph.D. student in the Department of Mechanical and Aerospace Engineering at the Universityof 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 inthe U.S. Army Reserve serving as a Cyber Operations Officer. His current research interests include cyber-physicalsystems, 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 ASPINLaboratory. He received a B.E. in Electrical Engineering from the Lebanese American University, an M.S. in Electricaland 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 NationalScience Foundation (NSF) Faculty Early Career Development Program (CAREER) award, and in 2019, he receivedthe Office of Naval Research (ONR) Young Investigator Program (YIP) award. He is a recipient of 2018 IEEEWalter Fried Award, 2018 Institute of Navigation (ION) Samuel Burka Award, and 2019 ION Col. Thomas ThurlowAward. He is an Associate Editor for the IEEE Transactions on Aerospace and Electronic Systems and the IEEETransactions on Intelligent Transportation Systems. His research interests include cyber-physical systems, estimationtheory, navigation systems, autonomous vehicles, and intelligent transportation systems.

ABSTRACT

An opportunistic navigation framework using low Earth orbit (LEO) satellites is analyzed. This framework, termedsimultaneous tracking and navigation (STAN), estimates a navigating vehicle’s state along with the states of orbitingLEO satellites. STAN employs an extended Kalman filter (EKF) to fuse measurements from satellite receivers and aninertial 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 receiverclock 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 atomicclock (CSAC). Additionally, the effect of utilizing a large number of LEO satellites for navigation is investigated. Thisanalysis provides insight into the achievable performance of STAN, which can serve as an alternative navigation systemin global navigation satellite system (GNSS)-denied environments. The performance predictions from simulations arecompared with experimental results with real signals from the Orbcomm LEO constellation. A close match betweenthe simulation and experimental results is demonstrated for an unmanned aerial vehicle (UAV) navigating via theSTAN framework with signals from two Orbcomm LEO satellites for 160 seconds, the last 35 seconds of which arewithout GNSS signals. The UAV’s position root-mean squared error (RMSE) from simulations was 8.6 m, while theexperimental position RMSE was 10 m.

I. INTRODUCTION

There has been a surge in recent years to establish resilient positioning, navigation, and timing (PNT) serviceswhich possess features of accessibility and integrity [1]. This surge embodies the paramount need for resilient PNTon 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 navigationpurposes in GNSS-denied environments.

Copyright c© 2020 by T. R. Mortlockand Z. M. Kassas

Preprint of the 2020 ION GNSS+ ConferenceSt. Louis, Missouri, September 21–25, 2020

The paper focuses in particular on unmanned aerial vehicles (UAVs), which traditionally navigate their trajectoriesby relying on a tightly coupled system of inertial measurement units (IMUs), used for short-term positioning updatesand a local navigation solution, and GNSS signals, used to correct accumulated errors from IMU measurements andto 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 availabilityof 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 ofexploiting 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 accuratenavigation has been demonstrated on ground vehicles with terrestrial cellular and television signals [22–24], whilesub-meter-level accurate navigation has been demonstrated on UAVs [25, 26]. Moreover, opportunistic navigationwith LEO signals has been demonstrated on ground vehicles and UAVs with existing constellations, while showingthe 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 arecurrently 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 of717 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 behalfof SpaceX to add 30,000 LEO satellites to their current efforts [30]. LEO satellite signals offer a number of uniquebenefits for navigation purposes: (i) strong signal strength due to their lower orbits, (ii) diversity in their geometriesand frequencies, (iii) availability from different orbits and constellations, and (iv) the ability to observe and collect freeof 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 assumedto be equipped with atomic oscillators, nor to be tightly synchronized, and (iii) extracting navigation observablesfrom LEO satellites is not yet fully understood. This paper focuses on the first two challenges by adopting thesimultaneous tracking and navigation (STAN) framework. The STAN framework estimates the dynamic, stochasticstates 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 LEOsatellite dynamical model in [20] and was adapted to account for the case where LEO satellites periodically transmittheir positions in [31]. More elaborate LEO satellite dynamics models were studied in [32]. A differential frameworkwas 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 frameworkusing 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 navigatingvehicle’s IMU grade and both the quality of the vehicle-mounted oscillator and LEO satellites’ oscillators on thenavigation solution is studied. Bounds for STAN’s performance as a function of the sensor types used, as well asthe number of LEO satellites used, are presented through simulation. Experimental results that utilize the STANframework for navigation is compared with the paper’s proposed performance characterization, showing a closematch between the simulation and experimental results. These results demonstrate a UAV navigating via the STANframework with signals from two Orbcomm LEO satellites for 160 seconds, the last 35 seconds of which are withoutGNSS signals. The UAV’s position root-mean squared error (RMSE) from simulations was 8.6 meters, while theexperimental position RMSE was 10 meters. The future of high-availability navigation requires overcoming bothpurposeful and incidental GNSS degradation alike, and the analysis of navigation with LEO satellites under theSTAN 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 characterizesthe 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 measurementsmade from an IMU, a GNSS receiver, and a LEO satellite receiver. A key difference between STAN and a traditionaltightly coupled GNSS-INS system is the estimation of the LEO satellites’ positions, velocities, and clocks along withthe vehicle’s states. During the prediction stage of the EKF; IMU, clock, and LEO propagation models are used topredict the filter’s state. Subsequently, during the update stage of the EKF, the filter uses measurements from theLEO 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 theLEO satellites. The filter states and the corresponding dynamics and measurement models of the filter are discussednext. Following this, an overview of LEO satellite megaconstellations and simulations using STAN are presented.

INS

LEO

IMU

State

Initialization

Orbit

Determination

Clock Models

EKF Prediction

EKF

GNSS

Receiver

LEO

Receiver

Propagation Update

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

A. STAN Models

The state vector of the EKF is defined as follows:

x =[

xT

r , xT

leo,1, . . . , xT

leo,M

]T

(1)

xr =[

BGq

T, rT

r , rT

r , bT

g , bT

a , cδtr, cδtr

]T

(2)

xleo,m =[

rT

leo,m, rT

leo,m, cδtleo,m, cδtleo,m

]T

(3)

where xr is the state vector of the vehicle consisting of BGq, 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 rr, 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 δtr, the clock bias and drift of the receiver, respectively;and c is the speed of light. The vector xleo,m is the state vector of the mth LEO satellite, consisting of rleo,m and

rleo,m, the 3-D satellite position and velocity, respectively; δtleo,m and δtleo,m, the satellite’s transceiver clock biasand 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 processedwith the strap-down INS kinematic equations [34]. The vehicle’s accelerometer and gyroscope biases, covered inSection III, are propagated according to a velocity random walk model. The clock states of the vehicle and LEOsatellites, 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 isthe second gravitational zonal coefficient [32]. During the EKF update, the vehicle-mounted LEO satellite receivermakes pseudorange and pseudorange rate measurements from each satellite. The Doppler frequency measurementsfD are made from the transmitted LEO satellite signals, from which a pseudorange rate measurement ρ can beobtained as ρ = − c

fcfD, where fc is the carrier frequency. The pseudorange measurement ρleo,m at time-step j from

the mth LEO satellite is modeled according to:

ρleo,m(j) = ‖rr(j)− rleo,m(j)‖2 + 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 effectscompared to the LEO satellite position and velocity estimate errors [35], and vρleo,m

is the measurement noise, whichis modeled as a white Gaussian random sequence with variance σ2

ρleo,m. The LEO receiver also makes pseudorange

rate measurements, ρleo,m, on the LEO satellites which are modeled following the same above assumptions as

ρleo,m(j) = [rleo,m(j)− rr(j)]T

[rr(j)− rleo,m(j)]

‖rr(j)− rleo,m(j)‖2+ c · [δtr(j)− δtleo,m(j)] + vρleo,m

(j), j = 1, 2, . . . , (5)

where vρleo,mis the measurement noise, which is modeled as a white Gaussian random sequence with variance σ2

ρleo,m.

Ultimately, the EKF outputs an estimate of the state vector, denoted x(k), and a corresponding estimation errorcovariance matrix, denoted P(k). The estimation error covariance matrix is an important representation of theachievable 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 theirrespective 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 ofLEO satellites. The specific constellation used in this section is the Starlink constellation of LEO satellites, operatedby SpaceX, which has already established a significant presence in LEO. The North American Aerospace DefenseCommand (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 satellitepositions was generated by pulling data from the NORAD web-server and utilizing a two-body with J2 perturbationsorbit propagation model to generate the satellite trajectories. This orbit model is a first-order approximation thatbuilds upon the traditional two-body model with the added J2 component. The general two-body model is writtenas [32]

rleo,m = agravm, agravm

=dUm

drleo,m, (6)

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 bewritten as [37]

Um =µ

‖rleo,m‖

[

1−

N∑

n=2

JnRn

E

‖rleo,m‖nPn (sin(θ))

]

, (7)

where Pn is a Legendre polynomial with harmonic n, Jn is the nth zonal coefficient, RE is the mean radius of theEarth, sin(θ) = zleo,m/‖rleo,m‖, rleo,m , [xleo,m, yleo,m, zleo,m]

Tare the position coordinates of the mth 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 June2020, SpaceX conducted more launches pushing their total to over 500 LEO satellites. Fig. 3(a) shows simulatedorbits of 118 of these active satellites that were visible over an elevation mask of 5◦ degrees for a stationary receiverat the University of California, Irvine (UCI) at some point during the satellites orbital period. The red sections ofthe orbits indicated when the satellites were visible over the elevation mask. Fig. 3(b) shows a heat map of thefuture 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 elevationangles, collection of signals from constellations like Starlink has not been studied, to the authors’ knowledge. In thisstudy, 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 ofvisible 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 ofLEO satellites from the Starlink megaconstellation. The LEO receiver was assumed to produce pseudorange andDoppler measurements to visible Starlink LEO satellites. The pseudorange and pseudorange rate measurement noisevariances 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 inperformance to a small private plane with a cruise speed of roughly 50 m/s. The UAV flies a 360-second trajectorycovering 21.8 km, shown in white in Fig. 4(a), consisting of a straight climbing segment, followed by a figure-eightpattern 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 backdown to 1 km in the straight segment. The LEO satellite states are initialized using TLE files and the trajectoriesof the 20 Starlink LEO satellites used for navigation are shown in white in Fig. 4(b). GNSS was available for thefirst 60 seconds of the flight, and STAN’s estimate for the satellites are displayed in green during this 60-second timeperiod. 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 thevehicle’s positioning accuracy [38]. For example, tactical, industrial, and consumer are all different grades of IMUs

(a)

(c)

(d)

(b)

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 scaleatomic clocks (CSAC) are three different types of clocks that have differing timing stability parameters, whichultimately 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 aspecific 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 measuresthe specific force acting on the vehicle, which in turn is orientated to the inertial frame, corrected for gravity, andintegrated twice to obtain the position estimate of the vehicle. There are various sensor errors that can distortaccelerometer and gyroscope measurements, namely, biases, random noises, and scale factor errors [39]. Propagationof these errors through the integrations described above has compounding effects on the degradation of the IMU’spositioning estimate [40]. This paper focuses on bias errors and the measurement noise statistics of the vehicle’s IMUwhile navigating with LEO satellites under STAN. The triad-gyroscope and triad-accelerometer produces angularrate ωimu and specific force aimu measurements, modeled as

ωimu(k) =Bω(k) + bg(k) + ng(k), k = 1, 2, . . . (8)

aimu(k) = R[

Bk

G q(k)]

(

Ga(k)− Gg(k))

+ ba(k) + na(k), k = 1, 2, . . . , (9)

where R[Bk

G q] is the rotation matrix representation of the quaternion vector from the global to the body frame, Gg

and Ga are the acceleration and the gravity acceleration in the global frame, and ng and na are measurement noise

vectors, which are modeled as white noise sequences with covariances σ2gI3×3 and σ2

aI3×3, respectively.

The gyroscope’s and accelerometer’s biases, bg and ba, respectively, are modeled to evolve according to a velocityrandom 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 withcovariances σ2

bgI3×3 and σ2baI3×3, respectively.

The following simulations study how different IMU biases and noise statistics affect the performance of a vehicleusing STAN for navigation during GNSS unavailability. The UAV’s trajectory in these simulations is fixed, whichis the trajectory detailed in Section II. The maneuvers a vehicle undergoes alter how the IMU errors propagateinto the navigation solution, and for this reason, the vehicle’s motion along the chosen trajectory excites all threedirections of both the accelerometer and gyroscope. The LEO satellites orbits are the same as those outlined inSection II, and are propagated using a two-body with J2 model. An elevation mask of 16.5◦ was set to maximizeLEO satellite availability for the cases of 0 to 50 LEO satellites. For the case of 100 LEO satellites, due to a lack oftotal visible satellites, an elevation angle of 10◦ was used with a combination of the current Starlink satellites andfuture 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 increasingthe number of LEO satellites used under the STAN framework. The UAV and LEO satellites were equipped with ahigh-quality TCXO and high-quality OCXOs, respectively, and GNSS was cut after 60 seconds. The IMU grade thevehicle was equipped with was varied from highest to lowest performance: tactical, industrial, and consumer. Duringeach simulation the biases and noise parameters for each IMU grade were altered to those listed in Table I. Note thatfor 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 coordinateframe, and the estimation error trajectories are not plotted in these figures in order not to convolute the plots. Eachcolor represents a different number of LEO satellites used for tracking and navigation, and appears on the plots threetimes for each of the different IMU grades. The dashed vertical line at 60 seconds on each plot represents the GNSScutoff time. The relatively sharp decrease around 160 seconds, most easily observed in the case of 5 LEO satelliteswith a consumer grade IMU in the north direction of Fig. 5, is due to the UAV’s trajectory. The wave-like increasingand decreasing values observed in the orientation states of the vehicle in Fig. 5 are attributed to the banking turns inthe vehicle’s trajectory. The following observations consider the first three plots, representing the uncertainty in thevehicle position states. As expected, after GNSS is cut off without any LEO satellites, the uncertainty diverges veryquickly. 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 areused, an industrial-grade IMU provides comparable performance to a tactical-grade IMU. Additionally, when 50 LEOsatellites are used, in the North position error, a consumer-grade IMU performs similarly to a tactical-grade IMUwith 20 LEO satellites. Furthermore, when 100 LEO satellites are used, a consumer-grade IMU provides roughly thesame performance as both the industrial-grade and tactical-grade IMUs. These results help determine achievablenavigation performance using STAN as a function of time without GNSS with different numbers of LEO satellitesand IMU grades.

TABLE I

IMU Grade Simulation Settings

Parameters Units Tactical Industrial Consumer

σg rad/s 2.036× 10−4 2.891× 10−4 5.236× 10−3

bg(0) rad/s 4.848× 10−6 2.417× 10−4 2.9× 10−3

σbg rad/s 1× 10−7 5× 10−7 1× 10−5

σa m/s2

1.629× 10−6 4.062× 10−6 2.452× 10−2

ba(0) m/s2 1.962× 10−3 2.943× 10−2 7.848× 10−1

σba m/s2

1× 10−7 5× 10−7 1× 10−5

GPS cutoff

East

[m]

North[m

]Up[m

]Up[m

/s]

East

[m/s]

North[m

/s]

500

-500

250

-250

0

500

-500

250

-250

0

500

-500

250

-250

0

0

10

-10

0

10

-10

0

10

-10

Roll[rad]

Pitch

[rad]

Yaw

[rad]

0

0.5

-0.5

0

0.5

-0.5

0

0.5

-0.5

Time [s]0 50 100 150 200 250 300 350

Number of LEO Satellites / IMU Grade0 / Tactical5 / Tactical20 / Tactical100 / Tactical

0 / Industrial5 / Industrial20 / Industrial100 / Industrial

0 / Consumer5 / Consumer20 / Consumer100 / Consumer

Fig. 5. EKF ±3σ estimation error bounds of the UAV’s states in the local navigation frame for varying IMU grades and number of LEOsatellites. 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 recentadvances 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 frequencyto that of the nominal, whereafter the power spectrum is determined. The noise power spectrum has been shownthrough 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 accordingto

xclk (k + 1) = Fclk xclk(k) +wclk(k), k = 1, 2, . . . , (12)

xclk ,

[

cδt, cδt]T

, Fclk =

[

1 T0 1

]

,

where T is the constant sampling interval and wclk is the process noise [47], which is modeled as a discrete-timewhite noise sequence with covariance

Qclk =

[

SwδtT + Sw

δt

T 3

3Sw

δt

T 2

2

Swδt

T 2

2Sw

δtT

]

. (13)

The terms Swδtand Sw

δtare the clock bias and drift process noise power spectra, respectively, which can be related

to the power-law coefficients, {hα,}2

α=0,−2, which characterizes the power spectral density of the fractional frequency

deviation of an oscillator from nominal frequency according to Swδt≈

h0

2and Sw

δt≈ 2π2h−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 usedoscillators, while also increasing the number of LEO satellites. Table II details the three different types of oscillatorsused in this section [48]. Fig. 6 shows the ±3σ estimation error bounds of the UAV while setting the LEO clocksas 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 numbersof LEO satellites, and each color is represented on the plots three times with a different line style for the varyingUAV oscillator quality. It is evident from Fig. 6 that the performance with the OCXOs are similar, while theTCXO diverges noticeably faster. With 100 LEO satellites, the performance with a TCXO becomes comparableto the OCXOs. Fig. 7 shows the effect of varying the LEO satellite clocks as CSACs, high-quality OCXOs, andtypical-quality OCXO while the UAV oscillator is held constant as a high-quality TCXO. An interesting finding isthat regardless of the number of LEO satellites used, the vehicles navigation performance is surprisingly not sensitiveto 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}

Typical-quality OCXO{

8.0× 10−20, 4.0× 10−23}

High-quality OCXO{

2.6× 10−22, 4.0× ·10−26}

CSAC{

7.2× 10−21, 2.7× 10−27}

GPS cutoff

East

[m]

North[m

]Up[m

]Up[m

/s]

East

[m/s]

North[m

/s]

500

-500

250

-250

0

500

-500

250

-250

0

500

-500

250

-250

0

0

10

-10

0

10

-10

0

10

-10

ClockBias[m

]ClockDrift

[m/s]

0

100

0

0.5Time [s]

0 50 100 150 200 250 300

Number of LEO Satellites / UAV Clock Type

0 / High-quality OCXO5 / High-quality OCXO20 / High-quality OCXO100 / High-quality OCXO

0 / Typical-quality OCXO5 / Typical-quality OCXO20 / Typical-quality OCXO100 / Typical-quality OCXO

0 / High-quality TCXO5 / High-quality TCXO20 / High-quality TCXO100 / High-quality TCXO

50

-50

-100

1

2

-2

-1

350

Fig. 6. EKF ±3σ estimation error bounds of UAV’s states in the local navigation frame for varying number of LEO satellites and UAVoscillator. The LEO satellite oscillators were fixed as high-quality OCXOs. The UAV was equipped with a tactical-grade IMU.

GPS cutoff

East

[m]

North[m

]Up[m

]Up[m

/s]

East

[m/s]

North[m

/s]

500

-500

250

-250

0

500

-500

250

-250

0

500

-500

250

-250

0

0

10

-10

0

10

-10

0

10

-10

ClockBias[m

]ClockDrift

[m/s]

0

500

0

0.5Time [s]

0 50 100 150 200 250 300

Number of LEO Satellites / LEO Satellite Clock Type

0 / CSAC5 / CSAC20 / CSAC100 / CSAC

0 / High-quality OCXO5 / High-quality OCXO20 / High-quality OCXO100 / High-quality OCXO

0 / Typical-quality OCXO5 / Typical-quality OCXO20 / Typical-quality OCXO100 / Typical-quality OCXO

250

-250

-500

2.5

5

-5

-2.5

350

Fig. 7. EKF ±3σ estimation error bounds of UAV’s states in the local navigation frame for varying number of LEO satellites and LEOsatellite 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 signalswere 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 EttusE312 was used to sample Orbcomm signals and store the in-phase and quadrature components. These samples werethen 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-basedestimator was used to implement the STAN algorithms for the UAV. Fig. 8 shows the experimental environment andnavigation results. To assess the fidelity of the simulation results presented in Section III, the experimental resultsare 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 thenavigation results obtained from the simulator versus the experiment. To the authors’ knowledge, these are the firstresults demonstrating navigation of a UAV with real LEO satellite signals, along with side-by-side comparison withhigh-fidelity simulations.

OrbcommLEO satellite 1trajectory

OrbcommLEO satellite 2trajectory

GPS cutoff

(a) (c)

(d)

Irvine, California

Flight start

Flight end

(b)

Final error: 18 m

Final error: 20 m

Experiment

Simulation

Truth

RMSE: 10 m

RMSE: 8.6 m

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

TABLE III

UAV Navigation Performance

Performance Measure Simulation Experiment

RMSE (m) 8.6 10Final Error (m) 18 20

V. CONCLUSION

This paper analyzed the achievable navigation performance of STAN with LEO satellites signals. Simulations ofactive Starlink satellites were used to navigate a fixed-wing UAV over a trajectory covering 21.8 km over 360seconds, the last 300 seconds of which was without GNSS signals. Different IMU grades were examined as a functionof the gyroscope and accelerometer used. It was demonstrated that increasing the number of LEO satellites usedcompensates 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 onnavigation 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 OrbcommLEO satellites for 160 seconds, the last 35 seconds of which are without GNSS signals. The experimental resultsshowed a close match with simulation results. The UAV’s position root-mean squared error (RMSE) from simulationswas 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 partby the National Science Foundation (NSF) under Grant 1929965. The authors would like to thank Joshua Moralesand Joe Khalife for helpful discussions.

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