Signals of Opportunity Aided Inertial NavigationJoshua J. Morales, Paul F. Roysdon, and Zaher M. Kassas

University of California, Riverside

BIOGRAPHIES

Joshua J. Morales is pursuing a Ph.D. from the Depart-ment of Electrical and Computer Engineering at the Uni-versity of California, Riverside. He received a B.S. inElectrical Engineering with High Honors from The Uni-versity of California, Riverside. In 2016 he was accordedan Honorable Mention from the National Science Founda-tion. His research interests include estimation, navigation,autonomous vehicles, and intelligent transportation sys-tems.

Paul F. Roysdon is pursuing Ph.D. and M.S. degrees fromthe Department of Electrical and Computer Engineeringat the University of California, Riverside. He earned aB.S. in Aeronautical and Mechanical Engineering from theUniversity of California, Davis, an M.S. in AeronauticalEngineering from the Von Karman Institute for Fluid Dy-namics, Belgium, and an M.S. in Mechanical Engineeringfrom the University of California, Davis. From 2005-2015he was a Research Engineer in the Advanced Programsdivision of Composite Engineering Inc. (CEi).

Zaher (Zak) M. Kassas is an assistant professor at theUniversity of California, Riverside and director of the Au-tonomous Systems Perception, Intelligence, and Naviga-tion (ASPIN) Laboratory. He received a B.E. in ElectricalEngineering from the Lebanese American University, anM.S. in Electrical and Computer Engineering from TheOhio State University, and an M.S.E. in Aerospace En-gineering and a Ph.D. in Electrical and Computer Engi-neering from The University of Texas at Austin. From2004 through 2010 he was a research and development en-gineer with the LabVIEW Control Design and DynamicalSystems Simulation Group at National Instruments Corp.His research interests include estimation, navigation, au-tonomous vehicles, and intelligent transportation systems.

ABSTRACT

A signal of opportunity (SOP)-aided inertial navigationsystem (INS) framework is presented and studied. Thefollowing problem is studied. A mobile receiver with ac-cess to global navigation satellite system (GNSS) signalsis aiding its onboard INS with GNSS pseudoranges. Whilenavigating, the receiver draws pseudorange observationson ambient unknown terrestrial SOPs and estimates theSOPs’ states. GNSS signals become unavailable, at whichpoint the receiver uses the SOPs to aid its INS. It is demon-strated that fusing SOP pseudoranges in a tightly-coupledGNSS-SOP-INS framework produces a superior naviga-

tion solution to a traditional tightly-coupled GNSS-INSframework. Moreover, in the absence of GNSS signals,it is demonstrated that aiding the INS via SOPs pro-duces bounded estimation errors. The SOP-aided INS’sperformance sensitivity is studied over varying quantityand quality of exploited SOPs. Experimental results arepresented demonstrating a ground vehicle navigating ex-clusively with inertial measurement unit (IMU) data andpseudoranges from an unknown terrestrial SOP emanatingfrom a cellular tower.

I. INTRODUCTION

Traditional navigation systems integrate global navigationsatellite system (GNSS) with an inertial navigation system(INS). When these systems are integrated, the long-termstability of a GNSS navigation solution complements theshort-term accuracy of an INS. However, it is well knownthat if GNSS signals become unavailable, the errors of anINS diverge. Recently, signals of opportunity (SOPs) havebeen considered to enable navigation whenever GNSS sig-nals become inaccessible or unreliable [1–6]. Future nav-igation systems could rely on SOPs to aid an INS in theabsence of GNSS signals, enabling a navigation solutionwith bounded errors.

Architectures to fuse GNSS and inertial measurementunit (IMU) measurements with loosely-coupled, tightly-coupled, and deeply-coupled estimators, are well-studied[7]. Regardless of the coupling type, the errors of a GNSS-aided INS will diverge in the absence of GNSS signals,and the rate of divergence depends on the quality of theIMU. Consumer and small-size applications that use af-fordable micro-electro-mechanical systems (MEMS) gradeIMUs are particularly susceptible to large error divergencerates. While high quality IMUs may reduce the rate of er-ror divergence, they may violate cost, size, weight, and/orpower constraints.

Current trends to supplement a navigation system in theevent that GNSS signals become unreliable are tradition-ally sensor-based (e.g., cameras [8], lasers [9], and sonar[10]). Recently, SOPs (e.g., AM/FM radio [11,12], cellular[13–17], digital television [18,19], iridium [20,21], and Wi-Fi [22, 23]) have been studied as a complement to GNSSor a stand-alone alternative. However, using SOPs as anaiding source for an INS received little attention. In [24],a board-mounted transceiver equipped with an IMU waspresented along with experimental results demonstratingthe use of SOP Doppler measurements to aid an INS. How-

Copyright c© 2016 by J.J. Morales, P.F. Roysdon,and Z.M. Kassas

Preprint of the 2016 ION GNSS+ ConferencePortland, OR, September 12–16, 2016

ever, details of the framework were obscure. This papercharacterizes the performance of a tightly-coupled frame-work that fuses IMU data with SOP pseudoranges alongwith GNSS pseudoranges (when available).

SOPs are transmitted at a wide range of frequencies and di-rections, making them an attractive supplement to GNSSsignals to improve the accuracy of a navigation solution[16, 17]. SOPs are abundant in GNSS-challenged environ-ments, making them particularly attractive aiding sourcesfor an INS when GNSS signals become unreliable. How-ever, unlike GNSS satellite vehicle (SV) states, the statesof SOPs, namely their position and clock states, may notbe known a priori and must be estimated. This estimationproblem is analogous to the simultaneous localization andmapping (SLAM) problem in robotics [25]. Both problemsask if it is possible for an agent to start at an unknown lo-cation in an unknown environment and then to incremen-tally build a map of the environment while simultaneouslylocalizing itself within this map. However, in contrast tothe static environmental map of the typical SLAM prob-lem, the SOP signal map is more complex– it is dynamicand stochastic. Specifically, for pseudorange-only observa-tions, one must estimate not only the position states, butalso the clock states of both the receiver and the SOPs[26, 27].

This paper considers the following practical problem. Amobile receiver, whether hand-held or vehicle-mounted,has access to GNSS SV observables, multiple unknownterrestrial SOPs, and IMU measurements, which are uti-lized to estimate the receiver’s states, (position, velocity,clock bias, and clock drift) and the SOPs’ states, (posi-tions, clock biases, and clock drifts). Suddenly, GNSSpseudoranges become unavailable, and the receiver con-tinues drawing pseudorange observations from the SOPtransmitters. The receiver continues navigating by fusingthe SOP pseudoranges with IMU measurements through adynamic estimator, which simultaneously maps the SOPs’states and localizes the receiver in that map using a pro-posed SOP-aided INS framework. This paper studies theperformance of the SOP-aided INS by addressing the fol-lowing two fundamental questions on uncertainty boundsof the receiver’s state estimates: (1) How are the uncer-tainty bounds affected by varying the number of exploitedSOPs? (2) How sensitive are the uncertainty bounds tothe the stability of the SOPs’ oscillators?

The remainder of this paper is organized as follows. Sec-tion II describes the dynamics model of the SOPs and nav-igating vehicle as well as the receiver’s observation model.Section III describes the SOP-aided INS framework andpresents simulation results. Section IV presents a perfor-mance sensitivity analysis of the SOP-aided INS frame-work over varying quantity and quality of exploited SOPs.Section V presents experimental results demonstrating aground vehicle navigating with cellular signals using the

SOP-aided INS framework. Concluding remarks are givenin Section VI.

II. MODEL DESCRIPTION

A. SOP Dynamics Model

Each SOP will be assumed to emanate from a spatially-stationary terrestrial transmitter, and its state vectorwill consist of its three-dimensional (3D) position states

rsop , [xsop, ysop, zsop]T

and clock states xclk,sop ,[

cδtsop, cδtsop

]T

, where c is the speed of light, δtsop is the

clock bias, and δtsop is the clock drift [28].

The SOP’s discretized dynamics are given by

xsop (k + 1) = Fsop xsop(k) +wsop(k), k = 1, 2, . . . ,

where xsop =[

rTsop, x

Tclk,sop

]T

, Fsop = diag [I3×3, Fclk],

wsop is the process noise, which is modeled as a discrete-time (DT) zero-mean white noise sequence with covarianceQsop = diag

[

03×3, c2Qclk,sop

]

, and

Fclk=

[

1 T

0 1

]

,Qclk,sop=

[

SwδtsopT+Swδtsop

T 3

3Swδtsop

T 2

2

Swδtsop

T 2

2Swδtsop

T

]

,

where T is the constant sampling interval. The termsSwδtsop

and Swδtsopare the clock bias and drift process

noise power spectra, respectively, which can be relatedto the power-law coefficients, {hα}

2

α=−2, which have beenshown through laboratory experiments to characterize thepower spectral density of the fractional frequency devia-tion of an oscillator from nominal frequency according toSwδtsop

≈h0

2and Swδtsop

≈ 2π2h−2 [29].

B. Vehicle Dynamics Model

The vehicle-mounted navigating receiver’s state vector xr

is comprised of the INS states xB and the receiver’s clock

states xclk,r ,

[

cδtr, c ˙δtr

]T

, i.e., xr =[

xTB, x

Tclk,r

]T

.

The INS 16-state vector is

xB =[

BGq

T, rTr , vT

r , bTg , bTa]T

,

where rr and vr are the 3-D position and velocity, respec-tively, of the body frame expressed in a global frame, e.g.,the Earth-centered Earth-fixed (ECEF) frame; bg and baare the gyroscope and accelerometer biases, respectively;and B

Gq is the 4-D unit quaternion in vector-scalar formwhich represents the orientation of the body frame withrespect to a global frame [30].

The orientation of the INS will evolve in DT according to

Bk+1

G q =Bk+1

Bkq ⊗ Bk

G q,

2

whereBk+1

Bkq represents the relative rotation in the body

frame from time-step k to k + 1, Bk

G q represents the ori-entation of the body frame in a global frame at time-stepk, and ⊗ is the quaternion multiplication operator. The

unit quaternionBk+1

Bkq is the solution to the differential

equation

Bt

Bk˙q =

1

2

[

−⌊Bω(t)×⌋ Bω(t)

−BωT(t) 0

]

Bt

Bkq, t ∈ [tk, tk+1],

where tk , kT . The vector Bω ,[

Bωx,Bωy,

Bωz

]Tis

the 3-D rotational rate vector, whereas the matrix ⌊Bω×⌋is the skew symmetric matrix representation of the vec-tor Bω. The velocity will evolve in time according to theintegration

vr(k + 1) = vr(k) +

∫ tk+1

tk

Ga(τ)dτ

where Ga is the 3-D acceleration of the IMU in the globalframe. The position will evolve in time according to theintegration

rr(k + 1) = rr(k) +

∫ tk+1

tk

vr(τ)dτ.

The evolution of bg and ba will be modeled as random

walk processes, i.e., ba = wa and bg = wg with E[wg] =E[wa] = 0, cov[wg] = σ2

wgI3, and cov[wa] = σ2

waI3. The

above attitude, position, and velocity models are discussedin [31].

The receiver’s clock states will evolve in time according to

xclk,r(k + 1) = Fclkxclk,r(k) +wclk(k),

where wclk is the process noise vector, which is modeledas a DT zero-mean white noise sequence with covarianceQclk,r, which is identical to Qclk,sop, except that Swδtsop

and Swδtsopare now replaced with receiver-specific spectra,

Swδtrand Swδtr

, respectively.

C. IMU Measurement Model

The IMU contains a triad-gyroscope and a triad-accelerometer which produce measurements zimu ,[

ωTimu, a

Timu

]T, which are modeled as

ωimu = Bω + bg + ng

aimu = R[Bk

G q](

Ga− Gg)

+ ba + na,

where R[q] is the equivalent rotation matrix of q, Gg isthe acceleration due to gravity in the global frame, and ng

and na are measurement noise vectors, which are modeledas zero-mean white noise sequences with covariances Qg

and Qa, respectively.

D. Receiver Observation Model

The pseudorange observation made by the receiver on themth SOP, after discretization and mild approximations dis-cussed in [28], is related to the receiver’s and SOPs’ statesby

zsopm(j) = ‖rr(j)− rsopm

‖2

+ c ·[

δtr(j)− δtsopm(j)

]

+ vsopm(j), (1)

where vsopmis modeled as a DT zero-mean white Gaus-

sian sequence with variance σ2sopm

. The pseudorange ob-

servation made by the receiver on the nth GNSS SV, aftercompensating for ionospheric and tropospheric delays isrelated to the receiver states by

zsvn(j) = ‖rr(j)− rsvn

(j)‖2

+ c · [δtr(j)− δtsvn(j)] + vsvn

(j), (2)

where, zsvn, z′svn

− cδtiono − cδttropo, δtiono and δttropoare the ionospheric and tropospheric delays, respectively,z′svn

is the uncorrected pseudorange, and vsvnis modeled

as a DT zero-mean white Gaussian sequence with varianceσ2svn

.

III. SOP-AIDED INERTIAL NAVIGATION

In this section, the SOP-aided INS framework is describedand simulation results are presented to demonstrate theframework’s performance in a practical scenario.

A. Problem Formulation

Consider a navigating vehicle, which is equipped with aGNSS-aided INS and a receiver capable of producing pseu-doranges to M unknown SOPs. The goal of the proposedSOP-aided INS framework is twofold. First, when GNSSpseudoranges are available, SOP pseudoranges will supple-ment the GNSS-aided INS to improve the accuracy of thenavigation solution. Second, when GNSS pseudorangesbecome unavailable, SOP pseudoranges will be used exclu-sively as an aiding source to correct the accumulating er-rors of the INS. To exploit SOPs for navigation, their statesmust be known [32–34]. However, in many practical sce-narios the SOP transmitter positions are unknown. Fur-thermore, the SOPs’ clock states are dynamic and stochas-tic; therefore, they must be continuously estimated. Totackle these problems, the SOP-aided INS framework willadapt a SLAM-type strategy that operates in a mappingmode when GNSS psuedoranges are available and a SLAMmode when GNSS pseudoranges are unavailable.

B. Framework

To correct INS errors using SOP pseudoranges, an ex-tended Kalman filter (EKF) framework similar to a tra-ditional tightly-coupled GNSS-INS integration strategy is

3

adopted, with the added complexity that the SOPs’ states,

denoted{

xsopm

}M

m=1, are simultaneously estimated along-

side the navigating vehicle’s states. To simplify the follow-ing discussion, consider a scenario where GNSS pseudor-anges are available during t ∈ [ 0, t0) (mapping mode) andunavailable during t ∈ [ t0,∞) (SLAM mode), where t0 isthe instant GNSS pseudoranges become unavailable.

B.1 EKF State Model

During the mapping mode, the EKF produces an estimatex(k|j) , E[x(k)|{z(i)}ji=1] of x(k), and an associated esti-

mation error covariancePx(k|j) , E[x(k|j)xT(k|j)] where

x ,

[

xTr , x

Tsop1

, . . . ,xTsopM

]T

, z ,[

zTsv, z

Tsop

]T

zsv = [zsv1, . . . , zsvN

]T, zsop =

[

zsop1, . . . , zsopM

]T,

k ≥ j, and j is the last time step an INS-aiding source wasavailable. The EKF error state is defined as

x ,

[

xTB, x

Tclk,r, r

Tsop1

, xTclk,sop1

, . . . , rTsopM

, xTclk,sopM

]T

,

(3)where

xB =[

θT rTr vT

r bTg bTa]T

,

where θ ∈ R3 is the 3-axis error angle vector. The posi-

tion, velocity, and clock errors are defined as the standardadditive error, e.g., rsop1

, rsop1− rsop1

. The orientationerror is related through the quaternion product

BGq =B

Gˆq ⊗ δq,

where the error quaternion δq is the small deviation of theestimate B

Gˆq from the true orientation B

Gq and is given by

δq =

[

1

2θT,

√

1−1

4θTθ

]T

,

During the SLAM mode, zsv is no longer available, there-fore, the receiver’s clock states xclk,r no longer need to

be estimated. However, the relative biases ∆δtm , δtr −δtsopm

, m = 1, ...,M , will now be estimated. Therefore,

at t0, the states xclk,r and{

xclk,sopm

}M

m=1are removed

from the estimator, and the relative clock state estimates∆xclkm are initialized as

∆xclkm =[

∆δtm,∆ˆδtm

]T

, m = 1, ...,M (4)

where

∆δtm , δtr − δtsopm, ∆

ˆδtm ,

ˆδtr −

ˆδtsopm

.

The estimation error of the new state vector x′ and thecorresponding estimation error covariance are initializedaccording to

x′ = Tx,

Px′ = TPxTT,

where T is the transformation matrix which maps x from(3) to

x′ =[

xTB, r

Ts1, ∆xT

clk1, . . . , rT

sM, ∆xT

clkM

]T.

A high-level diagram of this framework is illustrated inFig. 1.

Inertial Navigation

SystemEKFUpdate

Detector

EKF Prediction

GPS SOP

zimu

IMU

zsv zsop

PPS

data

P(j|j)

x(′)(k|j)

x(′)(j|j)

z

Receiver Receiver

P(k|j)

flag

Tightly-coupledT

Fig. 1. SOP-aided INS framework. The flag indicates if the SOP-

aided INS operates in the mapping mode: x(′)

≡ x and P ≡ Px; or

the SLAM mode: x(′)

≡ x′ and P ≡ P

x′ .

B.2 Propagation

Between aiding updates, the INS uses zimu and the dy-namics discussed in Section II to propagate the estimate(either x or x′) and produce the corresponding predictionerror covariance. During the mapping mode, the one-stepprediction error covariance is given by

Px(k + 1|j) = FPx(k|j)FT +Q, (5)

F , diag [ΦB, Fclk, Fsop, . . . , Fsop] ,

Q , diag[

QdB, c2Qclk,r, Qsop, . . . , Qsop

]

.

The propagation of xB and calculation of the DT lin-earized INS state transition matrix ΦB and process noisecovariance QdB are performed through standard INSequations as described in [35, 36].

During the SLAM mode, the prediction error covariancePx′(k + 1|j) has the same form as (5), except that F isreplaced with F′ , diag [ΦB, Fsop, . . . , Fsop] and Q is re-

placed with Q′ , TQTT. The process noise covarianceQclkm

of the initialized states (4) is readily shown to beQclkm

= Qclk,r +Qclk,sopm, m = 1, . . . ,M .

B.3 Update

When an INS-aiding source is available, the EKF updatestep will correct the INS and clock errors using the stan-dard EKF update equations [37]. In the mapping mode,

i.e., z ,[

zTsv, z

Tsop

]T, the corresponding Jacobian is

H =[

HTsv, H

Tsop

]T,

4

Hsv =

01×3 1Tsv1

01×9 hTclk 01×5M

......

......

...

01×3 1TsvN

01×9 hTclk 01×5M

,

Hsop=

01×3 1Tsop1

01×9 hTclk Hsop1

· · · 0

......

......

.... . .

...

01×3 1TsopM

01×9 hTclk 0 · · · HsopM

,

where 1svn,

rr−rsvn

‖rr−rsvn‖ , hclk , [1, 0]T, 1sopm

,

rr−rsopm

‖rr−rsopm‖ , and Hsopm

,

[

−1Tsopm

, −hTclk

]

. The update

will produce the posterior estimate x(j|j) and an associ-ated posterior estimation error covariance Px(j|j).

In the SLAM mode, only SOP pseudoranges are available,i.e., z = zsop. The adjusted measurement Jacobian is

H′=

01×3 1Tsop1

01×9 H′sop1

· · · 0

......

......

. . ....

01×3 1TsopM

01×9 0 · · · H′sopM

,

where H′sopm

,

[

−1Tsopm

, hTclk

]

. The update will produce

the posterior estimate x′(j|j) and an associated posteriorestimation error covariance Px′(j|j).

C. Simulation results

In this subsection the estimation performance of the SOP-aided INS framework is analyzed using a simulator whichgenerated (i) the “ground truth” states of the navigatingvehicle, (ii) the SOPs’ states, (iii) noise-corrupted IMUmeasurements of specific force and angular rates, and (iv)noise-corrupted pseudoranges to multiple SOPs and GPSSVs. Details of this simulator are provided next. Sub-sequently, estimation results produced by the SOP-aidedINS are compared to estimation results produced by a tra-ditional GPS-aided INS.

C.1 Simulator

The IMU signal generator models a triad gyroscope anda triad accelerometer. The data yi(t) for the ith axis ofthe gyroscope and accelerometer were generated at 100Hzaccording to

yi(t) = (1+ǫki)·[ui(t)+bi(t)+ηMAi

+ηQi+ηRRW i

+ηRRi],

where ui(t) is either the vehicle’s actual acceleration orangular rotation rate for axis i, ǫki

is the scale factor, bi(t)is a random bias which is driven by the bias instability,ηMAi

is the misalignment, ηQ is quantization noise, ηRRW i

is rate random walk, and ηRRiis rate ramp [38]. The

magnitude of these errors and their driving statistics aredetermined by the grade of the IMU. Data for a consumer-grade and tactical-grade IMU were generated for this work.

GPS L1 C/A pseudoranges were generated at 1 Hz ac-cording to (2) using SV orbits produced from Receiver In-dependent Exchange (RINEX) files downloaded on June1, 2016 from a Continuously Operating Reference Station(CORS) server [39]. SOP pseudoranges were generated at5 Hz according to (1) and the SOP dynamics discussed inSubsection II-B.

The simulated trajectory corresponded to an aerial vehi-cle and included two straight segments, a climb, and arepeating orbit, performed over a 200 second period. Thistrajectory was generated using a standard six degree offreedom (6DoF) kinematic model for airplanes [36]. Ex-cluding trajectories generated in a closed-loop fashion soto optimize the vehicle’s and SOPs’ estimates [40,41], thistype of open-loop trajectory has been demonstrated to pro-duce better estimates than an open-loop random trajec-tory [42, 43].

C.2 Results

Two estimation frameworks were employed to estimatethe states of the navigating vehicle: (i) the SOP-aidedINS equipped with a consumer-grade IMU and (ii) a tra-ditional tightly-coupled GPS-aided INS equipped with atactical-grade IMU. For both estimation frameworks GPSpseudoranges were set to be available for t ∈ [0, 100),and unavailable for t ∈ [100, 200]. The initial errorsof the navigating vehicle’s states were initialized accord-ing to xr(0|0) ∼ N [017×1, Pxr

(0|0)], where Pxr(0|0) ≡

diag[

(10−2) · I3×3, 9 · I3×3, I3×3, (10−4) · I6×6, 9, 1

]

. Forthe SOP-aided INS framework, the SOP state es-timates were initialized according to xsopm

(0|0) ∼N

[

xsopm(0),Psop(0|0)

]

, for m = 1, . . . ,M , where

xsopm(0) ≡

[

rTsopm

, 104, 10]T

, Psop(0|0) ≡ (104) ·

diag [1, 1, 1, 0.1, 0.01], and the positions {rsop,m}Mm=1

were surveyed from SOP cellular tower locations in down-town Los Angeles. The simulated trajectory, SOP posi-tions, and the position at which GPS was set to becomeunavailable are illustrated in Fig. 2.

Navigating vehicle’s trajectory SOP locations GPS cut off location

Fig. 2. True trajectory the aerial vehicle traversed (yellow) and SOPlocations (blue pins).

5

The resulting estimation error trajectories and correspond-ing 3σ bounds (blue) for the position, velocity, and atti-tude of the receiver and the position of one of the SOPs areplotted in Figs. 3 (a)-(l). The plots in Figs. 4 (a) and (b)correspond to the estimation errors of the receiver’s clockstates with GPS available and the plots in Figs. 4 (c)and (d) correspond to the estimation errors of one of theSOP’s clock states while GPS was available. Figs. 4 (e)and (f) correspond to the estimation errors of the relativebiases and drifts that were initialized when GPS becameunavailable, as was described in Subsection III-B.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Time [s] Time [s]

rr,N

[m]

rr,E[m

]rr,D

[m]

vN

[m/s]

vE[m

/s]

vD

[m/s]

θr[rad]

θp[rad]

θy[rad]

rsop,N[m

]rsop,E

[m]

rsop,D

[m]

GPS-aided INS: Error 3σ SOP-aided INS: Error 3σ GPS cut off

Fig. 3. The results of two simulations are illustrated. In both simu-lations, a navigating aerial vehicle had access to GPS pseudorangesfor only the first 100 seconds while traversing the trajectory illus-trated in Fig. 2. In the first simulation, a tightly-coupled GPS-INSintegration strategy using a tactical-grade IMU produced the estima-tion error trajectories and corresponding 3σ bounds (black). In thesecond simulation, the SOP-aided INS framework produced the esti-mation error trajectories and corresponding 3σ bounds (blue). Theerror and 3σ bound oscillations coincide with the repeating circularpath of the vehicle. North, East, and down (NED) errors are shownfor position and velocity. Roll, pitch, and yaw (rpy) errors are shownfor the orientation.

For a comparative analysis, the same navigating vehicleused a traditional tightly-coupled GPS-INS integrationstrategy with a tactical-grade IMU to estimate its states.The resulting position, velocity, and attitude estimationerror trajectories and corresponding 3σ bounds (black) areplotted in Figs. 3 (a)-(i).

(a) (b)

(c) (d)

(e) (f)

cδt r

[m]

cδt sop

[m]

c∆δt[m

]

c˜ δtr[m

/s]

c˜ δtsop[m

/s]

c∆δt[m

/s]

GPS-aided INS: Error 3σ SOP-aided INS: Error 3σ

Fig. 4. Estimation error trajectories and 3σ bounds for the clockstates of the SOP-aided INS framework (blue) and traditional GPS-aided INS (black). (a) and (b) correspond to the receiver’s clockstates while GPS was available and (c) and (d) correspond to theSOP’s clock states while GPS was available. (e) and (f) correspondto ∆xclk1 during the SLAM mode. The 3σ bound oscillations in (e)and (f) coincide with the repeating circular path of the vehicle.

The following may be concluded from these plots. First,when GPS pseudoranges become unavailable at t = 100seconds, the estimation error variances associated withthe traditional GPS-INS integration strategy begin to di-verge, as expected, whereas a bound can be establishedfor the errors associated with the SOP-aided INS. Second,the SOP-aided INS with a consumer-grade IMU almostalways yields lower estimation error variances when com-pared to the traditional GPS-INS integration strategy witha tactical-grade IMU.

IV. PERFORMANCE SENSITIVITY ANALY-

SIS

In the section, the performance sensitivity of the SOP-aided INS framework is studied for a varying quantity andquality of exploited SOPs.

A. Quantity of SOPs

To study the performance sensitivity of the SOP-aidedINS framework over a varying number of SOPs, the statesof the navigating vehicle, all available SOPs, and noise-corrupted measurements were generated using the simula-tor and settings described in Section III-C. Five separatesimulation runs were conducted. For each simulation, the

6

states of the navigating vehicle were estimated. The firstthree runs employed the SOP-aided INS with a consumer-grade IMU and M = 2, ..., 4 SOPs. The last two runs,employed a traditional tightly-coupled GPS-INS integra-tion strategy (M = 0) with (i) a tactical-grade IMU and(ii) a consumer-grade IMU. Fig. 5 illustrates the resultinglogarithm of the determinant of the estimation error co-variance for each run, which is proportional to the volumeof the estimation uncertainty ellipsoid [42]. Fig. 5 (a) cor-responds to the attitude (log {det [Pq]}), whereas Fig. 5(b) corresponds to the position (log {det [Prr

]}).

M = 4

M = 3

M = 2

M = 0

(a)

log

{

det

[

Pq

]}

(b)

log{det[P

rr]}

Time [s]

M = 0

GPS cut off

Fig. 5. An aerial navigating vehicle traverses the trajectory illus-trated in Fig. 2. GPS pseudoranges become unavailable at 100s(red dotted line) and the vehicle continues to navigate by fusing INSestimates and pseudoranges from M SOPs using the framework il-lustrated in Fig. 1. Figures (a) and (b) correspond to the logarithmof the determinant of the estimation error covariance for attitude,and position, respectively, for exploiting a varying number of SOPsin the navigating receiver’s vicinity. The two curves for M = 0 corre-spond to a traditional tightly-coupled GPS-aided INS equipped witha tactical-grade IMU (purple) and consumer-grade IMU (green).

The following may be concluded from these plots. First,the estimation uncertainties produced by the SOP-aidedINS associated with both B

Gq and rr are reduced whenM is increased. The sensitivity of the estimation uncer-tainty of B

Gq and rr to varying M is captured by the dis-tance between the log {det [Pq]} and log {det [Prr

]} tra-jectories, respectively. Second, although the SOP-aidedINS used a consumer-grade IMU, the resulting estimationuncertainty of rr for M = 2, . . . , 4 is always lower whencompared to the resulting estimation uncertainty producedby a traditional tightly-coupled GPS-INS using a tactical-grade IMU. Third, although the GPS-INS equipped witha tactical-grade IMU produces a lower estimation uncer-tainty of B

Gq while GPS is available, the uncertainty begins

to diverge when GPS becomes unavailable. This uncer-tainty becomes larger than the uncertainties produced bythe SOP-aided INS, for which a bound may be specifiedfor M = 2, . . . , 4.

B. Quality of SOPs

In this subsection, the performance sensitivity of the SOP-aided INS framework is studied over varying qualities ofSOPs. This work defines the SOP quality as the stabilityof the oscillator that is equipped on the SOP transmitter.The stability of the oscillator is characterized by the pa-rameters of Qclk,sop as described in Subsection II-A. Foursimulations were conducted using the simulator and set-tings described in Section III-C. The resulting 3σ boundsfor exploiting four SOPs, which were assumed to all beequipped with a worst temperature-compensated crystaloscillator (TCXO) (black), typical TCXO (green), typicaloven-controlled crystal oscillator (OCXO) (blue), and bestOCXO (purple), are plotted in Fig. 6. The four grades ofoscillators considered and their characterizing parametersare tabulated in Table I.

rr,N

[m]

Seconds [s]

(a)

(b)

(c)

(d)

(e)

(f)

rr,E

[m]

rr,D

[m]

vN[m

/s]

vE[m

/s]

vD[m

/s]

Worst TCXO Typical TCXO Typical OCXO Best OCXO

GPS cut off

Fig. 6. An aerial navigating vehicle traverses the trajectory illus-trated in Fig. 2. GPS pseudoranges become unavailable at 100s (reddotted line) and the vehicle continues to navigate using the frame-work illustrated in Fig. 1. Figures (a)-(f) correspond to the 3σbounds for exploiting SOPs equipped with a worst TCXO (black),typical TCXO (green), typical OCXO (blue), and best OCXO (pur-ple). North, East, and up (NED) position and velocity errors areshown for the worst TCXO (red) and best OCXO (orange).

From these plots it may be concluded that sensitivity ofthe estimation performance to the quality of oscillator wasminimal while GPS was available. When GPS pseudor-anges are unavailable, the estimation performance is sig-nificantly more sensitive to the quality of the oscillator,

7

TABLE I

Quality of SOPs: Oscillator Type

Parameter Value

Worst TCXO {h0,s, h−2,s}{

2.0× 10−19, 2.0× 10−20}

Typical TCXO {h0,s, h−2,s}{

9.4× 10−20, 3.8× 10−21}

Typical OCXO {h0,s, h−2,s}{

8.0× 10−20, 4.0× 10−23}

Best OCXO {h0,s, h−2,s}{

2.6× 10−22, 4.0× 10−26}

and the sensitivity is captured by the distance betweenthe 3σ trajectories. Although the uncertainty in the esti-mates were larger when SOPs were equipped with a worstTCXO, a bound may still be established.

V. EXPERIMENTAL RESULTS

A field experiment was conducted using an IMU and a soft-ware defined receiver (SDR) to demonstrate the SOP-aidedINS framework discussed in Section III-B. To this end, twoantennas were mounted on a vehicle to acquire and trackmultiple GPS signals and a cellular base transceiver sta-tion (BTS) whose signals were modulated through codedivision multiple access (CDMA). The GPS and cellularsignals were simultaneously downmixed and synchronouslysampled via a two-channel National Instruments R© univer-sal software radio peripheral (USRP). This front-end fedthe data to the Multichannel Adaptive TRansceiver Infor-mation eXtractor (MATRIX) SDR, which produced pseu-dorange observables from all GPS L1 C/A signals in view,and the cellular BTS [14]. The IMU data was sampledfrom a navigation system developed at the University ofCalifornia, Riverside (UCR), which is equipped with: (i) aconsumer-grade IMU, (ii) a tactical-grade IMU, and (iii)a u-blox R© GPS receiver. Fig. 7 depicts the experimentalhardware setup.

Experimental results are presented for two estimators:(i) the proposed SOP-aided INS using the consumer-grade IMU and (ii) for comparative analysis, a traditionaltightly-coupled GPS-INS using the consumer-grade IMU.For both estimators, GPS pseudoranges were available foronly the first 16 seconds of the 30 second run. The groundtruth trajectory was generated with a common GPS-INSEKF using the u-blox GPS receiver and the tactical-gradeIMU. Fig. 8 is an illustration of the true and estimated ve-hicle trajectories, the true and estimated tower locations,and the North-East 99th-percentile initial and final uncer-tainty ellipses. The North-East root mean squared error(RMSE) of the traditional tightly-coupled GPS-INS’s navi-gation solution after GPS became unavailable was 23.5 me-ters. The SOP-aided INS produced a trajectory estimatewith an RMSE of 9.42 meters and a final BTS localizationerror of 15.5 meters. Ir is worth noting that for longer pe-riods of GPS unavailability, the RMSE reduction from the

SOP-aided INS will be even more significant, since the er-rors of the unaided INS will diverge, while the SOP-aidedINS errors are expected to remain bounded.

MATLAB-BasedFilter

GPS and cellular antennas

Universal software radio peripheral (USRP)

MATRIXLabVIEW-Based SDR

UCR Nav. System

Fig. 7. Experiment hardware setup

Estimated tower True towerlocationlocation

Final and initialuncertainty ellipses

True (GPS)

INS only

SOP-aided INS

Trajectories

GPS cut offlocation

Fig. 8. Experimental results

VI. CONCLUSION

This paper presented and studied an SOP-aided INSframework. The performance of the framework was com-pared against a traditional tightly-coupled GNSS-INS inte-gration strategy and the performance sensitivity was stud-ied by varying the quantity and quality of exploited SOPs.It was demonstrated that a bound could be establishedon the estimation errors in the absence of GNSS. TheSOP-aided INS using a consumer-grade IMU was shownto produced estimation uncertainties lower than a tra-ditional tightly-coupled GNSS-INS using a tactical-gradeIMU when two, three, or four SOPs were exploited. Fur-thermore, it was shown that SOPs equipped with low-quality oscillators may serve as effective INS-aiding sourcesto establish a bound on INS errors in the absence of

8

GNSS. Moreover, experimental results demonstrated a ve-hicle navigating with the SOP-aided INS framework in theabsence of GNSS, which yielded an RMSE reduction of59.9% when compared to an unaided INS.

Acknowledgment

This work was supported in part by the Office of NavalResearch (ONR) under Grant N00014-16-1-2305. The au-thors would like to thank Jesse Garcia for his help withdata collection.

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