Multiple GPS Fault Detection and IsolationUsing a Graph-SLAM Framework

Sriramya Bhamidipati, Student Member, ION and Grace Xingxin Gao, Senior Member, IONUniversity of Illinois at Urbana-Champaign

BIOGRAPHIES

Sriramya Bhamidipati is a Ph.D. student in the Department of Aerospace Engineering at the University of Illinois at Urbana-Champaign, where she also received her master’s degree in 2017. She obtained her B.Tech. in Aerospace from the IndianInstitute of Technology Bombay in 2015. Her research is related to developing robust and attack-resilient PNT solutions withapplications to power systems and UAVs using signal processing algorithms.

Grace Xingxin Gao received the B.S. degree in Mechanical Engineering and the M.S. degree in electrical engineering fromTsinghua University, Beijing, China in 2001 and 2003. She received the PhD degree in Electrical Engineering from StanfordUniversity in 2008. From 2008 to 2012, she was a research associate at Stanford University. Since 2012, she has been withUniversity of Illinois at Urbana-Champaign, where she is presently an assistant professor in the Aerospace Engineering.

ABSTRACT

Autonomous vehicles operating in GNSS challenged urban environments are prone to uncorrelated multipath effects in multiplesatellite channels. In addition, the increasing number of GNSS satellites due to the addition of new constellations increases theprobability of multiple satellite faults caused by broadcast anomalies. We propose a graph-based Simultaneous Localizationand Mapping (Graph-SLAM) framework to perform multiple GPS Fault Detection and Isolation (FDI), in particular, satellitefaults due to broadcast anomalies and received signal faults due to multipath.

SLAM is a well-known technnique in robotics. It utilizes sensor measurements to estimate the landmark features in anunknown 3-Dimensional (3D) map while simultaneously localizing the robot within it. Analogous to this, we design a Graph-SLAM framework, where the robot is the receiver, and the landmarks in the map are the GPS satellites. Utilizing the pseudo-ranges, receiver and satellite motion model, our SLAM-based FDI simultaneously estimates the position, velocity and time ofboth GPS receiver and satellites. Thereafter, we assess the probability of fault in each satellite by individually evaluating thecorresponding test statistic against its empirical cumulative distribution calculated on-the-fly.

We validate our algorithm via different experimental scenarios, namely, adding multiple simulated broadcast anomalies tothe open-sky data collected using a ground vehicle; flying an aerial vehicle in an urban area prone to multipath. We demonstratethe capability of our algorithm in performing multiple FDI while accurately locating the receiver.

1 INTRODUCTION

Recently, there has been a wide-spread utilization of GNSS-based navigation system for autonomous applications [1] such asself-driving ground vehicles [2] and Unmanned Aerial Vehicles [3]. Assessing the probability of GNSS satellite fault is a well-formulated framework in the context of aircraft navigation. However, autonomous vehicles operate in unpredictable conditions,which leads to additional challenges [4]. Some of the major challenges described in [5] include static infrastructure such asbuildings and thick foliage as well as dynamic obstacles such as traffic and pedestrians.

In particular, this work is focused on two major faults experienced by Autonomous Vehicles (AVs), namely, satellite andreceived signal faults. Satellite faults [6] occur due to the anomalies in different GNSS segments i.e., receiver malfunctionin user segment, clock anomalies in space segment and satellite broadcast anomalies due to control segment. The space andcontrol segment based anomalies require a few hours to be rectified due to the lack of satellite visibility from the ground station.Some of the most recent real-world incidents reporting satellite faults are listed as follows: non-standard codes were transmitted

by a satellite (SV49) [7] in 2017 and wrong timing offsets were broadcast by multiple satellites [8] in 2016. In addition, detaileddocumentation regarding various satellite faults that have occurred in the past decade are listed in the literature [9].

Received signal faults occur due to the presence of dense tall buildings in urban areas which cause satellite blockage andmultipath effects in multiple satellites. Recently, a drone inspecting the Millenium Tower in San Fransisco dropped from thesky due to the loss of GPS signal [10]. Unlike satellite faults, these received signal faults are more frequent and therefore,require constant monitoring. Urban navigation and the increasing number of GNSS satellites due to multiple constellationssuch as GPS, GLONASS, Galileo and Beidou also increases the probability of multiple faults [11].

Traditional Multiple Hypothesis Solution Separation (MHSS)-based RAIM algorithm utilize ranging information fromredundant (> 4) satellite signals to assess the receiver integrity [12]. Considering a prior hypothesis of having multiple faultymeasurements simultaneously, MHSS RAIM evaluates the separation between position estimated by considering all satellitesin view and position computed by excluding the hypothesized faulty satellites. However, with the increase in number ofsatellites, the implementation of MHSS RAIM becomes quite computationally expensive. While these traditional methodsshow promising results for applications in civil aviation, they are not directly applicable for AVs that navigate in urban areaswith limited number of non-blocked satellites and the presence of uncorrelated multipath in multiple satellite channels. There isa need for reliable framework to detect and isolate multiple satellite and/or received signal faults associated with each satellite.

We propose to perform multiple GPS Fault Detection and Isolation (FDI) by utilizing Simultaneous Localization and Map-ping (SLAM) framework. Our SLAM-based FDI not assesses GPS faults associated with each satellite individually but alsoeffectively isolates them to accurately estimate the Position, Velocity and Time (PVT) of the receiver.

The rest of the paper is organized as follows: Section II describes our SLAM-based FDI algorithm and its key characteristics;Section III experimentally validates the performance of our algorithm in detecting and isolating multiple faults while accuratelyestimating the receiver location; Section IV concludes the paper.

2 SLAM-BASED FDI

In this section, we provide an outline of our architecture and later explain the algorithmic details and initialization conditions.

Figure 1: Our proposed Grpah-SLAM framework detects and isolates multiple faults caused by satellite broadcast anomaliesand multipath effects. Orange stars denote the GPS satellites which are considered as landmarks. Blue triangles indicate thetrajectory of GPS receiver estimated using our SLAM-based FDI and gray triangles indicate trajectory of the GPS receiverpropagated via motion model.

SLAM [13] is a well-known technique in robotics. It utilizes sensor measurements to estimate the landmarks in an un-known three-dimensional (3D) map while simultaneously localizing the robot within it. Analogous to this, we design a Graph-SLAM [14] framework, where the robot is our GPS receiver and the GPS satellites are considered as landmarks in the map. Weaim to simultaneously update the PVT of receiver and landmarks in the map i.e., satellites. Our algorithm utilizes sequentialmeasurements to tackle multiple faults without increasing the computational complexity. We define fault as the probability oflarge unacceptable errors associated with each satellite caused by broadcast anomalies and multipath.

2.1 Our Architecture

We propose our SLAM-based FDI architecture, shown in Fig. 1, to detect and isolate multiple faults. The details are describedas follows

1. Firstly, we initialize our graph using PVT of the receiver and satellites computed via existing GPS algorithms [15].

2. We obtain measurements from different modules which include receiver motion model, satellite orbital model and pseu-doranges corresponding to the GPS satellites in view. These measurements are provided to Graph-SLAM module.

3. In Graph-SLAM module, we optimize the total error using an M-estimator-based cost function. The total error com-prises of terms that represent the error in GPS measurement model, receiver and satellite dynamics weighted by thecorresponding fault probability computed at the previous time instant.

4. Next, we execute optimization in two stages: one is tracking which minimizes the cost function on a couple of the mostrecent time instants; the other is mapping which we perform periodically, to globally minimize the cost function acrossall time instants. The outputs from this module are the PVT of receiver and satellites.

5. For each satellite, we individually compute the fault probability by evaluating the test statistic against the empiricalcumulative distribution computed on-the-fly.

6. Based on the fault probability associated with each satellite, we repeat the above steps to iteratively build the graph bysimultaneously localizing the PVT of receiver and satellites.

2.2 Input Measurements

The measurements provided as input to our Graph-SLAM are explained as follows

2.2.1 GPS Signal Model

The digitized baseband ith satellite siτ composed of a low-rate navigation bit sequence Di·, a unique L1 C/A codesequence Gi· with a chipping rate of fC/A = 1.023 MHz and a residual sinusoidal carrier sequence exp· is given by

siτ , P iτU iτ − τ id

exp

j2πf iDτ + φi

(1)

where U i· = Di·Gi·, τ is a vector of sampling times 0, 1fs, 2fs, ..., ∆T − 1

fs within a time interval ∆T , sampling

frequency of fs and P i denotes the received signal power. f iD denotes the carrier Doppler frequency, τ id denotes the code delayand φi represents the carrier phase.

By tracking the GPS satellite signal parameters namely τ id, fiD, φ

i using existing algorithms [15], we obtain the transmittime corresponding to each satellite signal. Thereafter, at time instant t, we compute pseudoranges ρi as

ρi = ‖pi − p‖+(cδt− cδti

)+ cδtiatmos + cδtimp + ηi

ρit = h(Xt, Yit)(2)

where c denotes the speed of light, p = [x, y, z], p = [x, y, z], cδt and cδt are the receiver 3D position, 3D velocity, clock biasand clock drift respectively such that Xt = [p, cδt, p, cδt]t. Similarly, Yit = [pi, cδti, pi, cδt

i]t are the ith satellite position,

velocity, clock bias and clock drift corrections respectively. In addition, cδtiatmos denote the atmospheric errors caused dueto ionospheric and tropospheric effects, cδtimp represents the errors induced due to the presence of multipath (if any) and ηi

represents the additive white Gaussian noise (AWGN).

2.2.2 Receiver Motion Model

We consider a constant velocity motion model g(.) and address the vehicle accelerations using a dynamic process noise covari-ance Q given by

Xt = g(uR,t,Xt−1,Σt−1) (3)

where Xt = AXt−1 and Σt = AΣt−1AT +Qt−1 are the receiver state and covariance matrix at t time instant predicted by the

receiver motion model. uR,t denotes the control input by the vehicle on which the receiver is mounted to maintain a constantvelocity motion. Also, σ = [σp, σp, σp, σ ˙cδt] and σp = 1 + 250/(min(max(‖p‖2, 52), 252))

A =

[I ∆T0 I

]Qt = A

[0 00 σ

]AT .

The dynamic process noise model captures the intuition that it is more difficult for a vehicle’s velocity to change whenits speed is large as compared to when its speed is small. Thus, we determine the coefficients of σp from a least squares fit toacceleration time data of a generic vehicle [16]. We obtain the Allan deviation σcδt of the receiver clock from the manufacturer’sdatasheet.

2.2.3 Satellite Orbital Dynamics Model

The satellite orbital dynamic model f(.) estimates the position, velocity and clock corrections of each ith satellite. This iscomputed using the orbital parameters and their corrections obtained by decoding the transmitted navigation message [17].

2.3 Our Graph Formulation

Each layer in our graphical framework consists of units called “nodes”, as seen in Fig. 2. The orange nodes represent themeasurements received whereas the blue nodes indicate the unknown states to be estimated. The functionality of various nodesare described as follows

1. The blue nodes in the first layer depict the sequential time series of the receiver’s PVT X1:t.

2. Similarly, each sub-layer in the second layer consists of blue nodes which represent the sequential PVT time series ofsatellites in view Yi

1:t ∀i ∈ 1, . . . , N.

3. Both the first and second layers are collectively constrained by GPS pseudoranges depicted by ρit.

4. At any time instant, the receiver node is constrained by its motion model uR,1:t and the individual satellite nodes areconstrained by their satellite orbital model ui1:t decoded from the corresponding navigation message transmitted by theith satellite.

5. In addition, each satellite node Yit is connected to a corresponding fault node Iit−1 with values ranging between 0 − 1,

where 0 indicates low and 1 indicates high fault probability associated with ith satellite.

Figure 2: Framework of our Graph-SLAM module. The three measurement nodes, indicated in orange, correspond to thereceiver motion model uR,t, satellite orbital dynamic model uit and pseudoranges ρit. The state nodes to be estimated, indicatedin blue, represent the sequential time series of the receiver X1:t, satellites Y1:N

1:t and satellite fault probability Iit .

2.4 Graph-SLAM Module

In our Graph-SLAM module, we execute optimization in two stages: one is tracking and the other is mapping. In tracking, ateach instant, we optimize the cost function on a couple of most recent time instants to obtain the corresponding PVT of boththe receiver and satellites. In the back-end mapping, we perform global graph optimization by minimizing the cost functionover all the time instants.

2.4.1 Error-based cost function

Different proposed SLAM algorithms [18] in robotics rely on minimizing the least-square residual-based cost function bymodeling these errors as a Gaussian distribution. However, during GPS satellite faults, the errors in outlier measurementsexhibit large tails that no longer follow a Gaussian distribution. Therefore, we utilize a combination of Huber M-estimator [19]and a robust redescending M-estimator known as the Tukey bisquare M-estimator [20] to formulate the cost function to beminimized.

The bisquare M-estimator ΛB(x,R), given in Eq. (4) totally rejects the gross outliers while not fully ignoring the moderatelylarge errors. However, this estimator is not convex because of which only local minimum exists. This leads to difficulties inconvergence if the initial state is far away from global minimum.

ΛB(x,R) =

(R−1x

)(1−

(1− (

(R−1x

)kB

)2)3) |x| ≤ kB

0 otherwise(4)

ΛH(x,R) =

0.5(R−1x

)2 |x| ≤ kHkH

(|(R−1x)| − kH

)otherwise

(5)

In Eq. (4), kB denotes the bisquare constant and in Eq. (5), kH denotes the huber constant. In addition, x denotes theresidual and R denotes the covariance associated with the residual.

In contrast, the Huber M-estimator ΛH(x,R) exhibits global convergence as seen in Eq. (5). However, they have loweraccuracy than biquare estimators because they assign equal weights to all the measurement residuals.

To account for the above-mentioned limitations, we switch between the two estimators based on different conditions. Duringinitialization and when the number of satellites ≤ 4 we utilize Huber M-estimator Λ = ΛH to ensure convergence; in all theother conditions, we opt for bisquare M-estimator Λ = ΛB to achieve better accuracy.

2.4.2 Tracking

In the tracking, we optimize the graph at each time instant, using the M-estimator-based cost function et which consists of threecomponents [21].

et(Xt, Y1:Nt ) =

∑i

Λ((ρit − h(Xt,Yi,t)

), Iit−1 + Ωit

)+ Λ

((Xt − g(uR,t,Xt−1,Σt−1)

), Σt

)+∑i

Λ((

Yit − f(uit)

), Iit−1

) (6)

Xt−W :t, Y1:Nt−W :t = argmin

(t∑

n=t−Wen(Xn, Y

1:Nn )

)(7)

In Eq. (6), Ωit denote the covariances of the measured pseudoranges of the ith satellite. The first term in the loss function et,given in Eq. (7), denotes the summation of residual errors between the pseudorange measurements ρit and that computed fromthe GPS signal model h(.) across the satellites in view. In the second term, we compare the receiver state to be optimized withthat of the receiver motion model g(.). In the third term, we compare the satellite position to be optimized with that of thesatellite orbital dynamics model f(.) deoded from the broadcast navigation message. Thereafter, we carry out optimization onthe most recent W measurements to estimate the variables Xt−W :t, Y

1:Nt−W :t.

2.4.3 Mapping

The mapping step is performed to bind the overall drift errors and correct the PVT of receiver and satellites based on global mapgenerated. This occurs at a slower update rate as compared to tracking so as to reduce the overall computational complexity. Weexecute mapping thread to constrain the drifts and obtain globally corrected estimates of the receiver X1:t and satellites Y1:N

1:t

given by

(X1:t, Y1:N1:t ) = argmin

(t∑

n=1

en(Xn, Y1:Nn )

)(8)

2.5 Multiple FDI

To detect and isolate multiple faults, we individually compute the test statistic γi for each satellite, that comprises of two terms:one is the difference between expected pseudorange computed using our Graph-SLAM and measured pseudorange; the other isdifference in PVT of satellite estimated using our Graph-SLAM and that obtained using the broadcast ephemeris decoded fromthe navigation message, i.e., γi = ||Yi

t − f(uit)||+ |ρit − h(Xt, Yit)|.

In the absence of faults, the values of this test statistic γi lies on a multivariate Gaussian distribution Γi ∼ N (.) based onour GPS signal model [17]. Therefore, we formulate our multiple FDI as follows

H0 : Γi ∼ N (µi, ωi)

H1 : Γi ∼ si(θ) +N (µi, ωi)(9)

where mean µi and covariance ωi are the parameters of Gaussian receiver-independent error distribution. s(θ) denotes the errorinduced by fault source θ such as multipath, broadcast anomaly and so on. H0 denotes the null hypothesis that there is no faultassociated with ith satellite whereas H1 denotes the alternate hypothesis that there is a fault.

To evaluate this, we consider the most recent M test statistic values and perform an on-the-fly empirical estimation tocompute the values of mean µit = mean(γij) and covariance ωi = std(γij) ∀j ∈ t− 1, . . . , t−M. Later, at any time instant,for each GPS satellite, we evaluate the cumulative distribution function of Γi evaluated at γit which is given by

ΦiΓ,t(γit) = P (Γ < γit |) (10)

Intuitively, in the presence of fault, our test statistic would lie in the low probability areas of the Gaussian distribution. Inour work, we consider 25% area towards the tails of Gaussian distribution to be high fault regions. Based on this, we estimateour probability of fault as

Iit =

4(

ΦiΓ,t(γit)− 0.5

)2

if 0.25 < ΦiΓ,t(γit) < 0.75

1 otherwise(11)

Later, if the value of Iit < 0.55, we utilize its corresponding γit to update the mean µit+1 and covariance ωit+1 for the nexttime instant. During initialization, we estimate the parameters of Gaussian error distribution i.e., mean µi0 and covariance ωi0from the incoming data (if reliable) or from other existing open-source non-faulty GPS data. In addition, we set the initial valueof our satellite fault nodes to I1:N

0 = 0.5 indicating neutrality.

Figure 3: Data collection setup consisting of a USRP device triggered by a CSAC. The raw GPS signals are logged at a samplingrate of 5 MHz.

3 EXPERIMENTS

We validate our proposed SLAM-based FDI algorithm in detecting and isolating multiple faults caused due to broadcastanomaly and multipath effects. We also analyze the fault probability associated with each satellite and demonstrate the in-creased robustness of our algorithm to localize the receiver in the presence of faults.

Our experimental setup is equipped with an AntCom 3GNSSA4-XT-1 GNSS antenna. We collected raw GPS samples ata sampling rate of 5 MHz using Universal Software Radio Peripheral (USRP-N210) equipped with a DBSRX2 daughterboardand connected to an external Microsemi Quantum SA.45s CSAC as shown in Fig. 3. We considered the constants kH = 4.5 inHuber M-estimator and kB = 3.9 in Bisquare M-estimator.

3.1 Experiment 1: Multiple Satellite Broadcast Anomalies

Our first set of experiments are conducted on a moving ground vehicle under open-sky conditions, with 6 satellites in view.During the interval t = 15 s to t = 40 s, we added simulated broadcast anomalies to 3 satellites in view, namely, PRN 7, 11and 28 thereby inducing positioning errors of 3 km, 13 km and 6 km respectively. This is achieved by manipulating the orbitalparameters obtained from the decoded navigation message. Given 4 satellites are required to estimate navigation solution andthere are 6 visible satellites, the conventional MHSS RAIM) [12] described in Section 1, fails to detect and isolate all the faultysatellites.

Figure 4: The fault probability Iit associated with ith satellite is shown for 4 satellites, of which PRN 1 is non-faulty and theothers i.e., PRN 7, 11 and 28 suffer from satellite broadcast anomalies during the interval t = 15 s to t = 40 s. Our SLAM-based FDI algorithm accurately detects the presence of anomalies which is observed via increase in the faulty probabilityabove Iit > 0.5, as indicated by the red line.

Utilizing our SLAM-based FDI algorithm, we observed that the fault probability Iit , seen in Fig. 4, increases from an RMSEof 0.28 to 0.57 for PRN 7, 0.36 to 0.81 for PRN 11 and 0.19 to 0.61 for PRN 28 during the presence of fault. In addition,our algorithm also accurately estimates that PRN 1 remains non-faulty at all times with an RMSE fault probability of 0.22.Therefore, we demonstrated that our algorithm not only detects and isolates multiple faults but also accurately estimates lowfault probability for the non-faulty satellites.

(a) UAV (b) Flat-bed truck

Figure 5: Experiment conducted in Champaign, Illinois using an aerial vehicle shown in (a), by flying it inside the netted cagemounted on the back of a flat-bed truck shown in (b). The experimental data captured the 3D dynamics of the UAV and GPSmultipath caused by tall buildings.

3.2 Experiment 2: Multipath Effects

We conducted our next set of experiments in Champaign, Illinois by mounting a GPS antenna on the aerial vehicle shown inFig. 5(a). To comply with the FAA regulations, we flew the UAV inside a netted cage mounted on the back of a flat-bed truckshown in Fig. 5(b). In urban areas, our SLAM-based FDI algorithm assesses the fault probability associated with each satelliteand also adaptively utilizes this information to simultaneously localize both the receiver and satellites in view.

(a) At t = 60 s (b) t = 74 s

Figure 6: Snapshots at t = 60 s and t = 74 s, which indicate the satellite fault probability computed using our SLAM-basedFDI algorithm for 4 satellites namely PRN 8, 9, 22 and 26. Green nodes indicate low probability of fault, i.e., Iit ≤ 0.5 and rednodes indicate high probability, i.e., Iit > 0.5. When the receiver is surrounded by buildings, our SLAM-based FDI accuratelydetects the 3 satellites prone to multipath.

In Fig. 6, we showed two snapshots at t = 60 s and t = 74 s, which indicate the probability of fault in each satellite,computed using our SLAM-based FDI algorithm. We plotted 4 out of 8 visible satellites, namely, PRN 8, 9, 22 and 26, withgreen indicating Iit < 0.5, i.e., low fault probability and red indicating Iit > 0.5, i.e., high probability. At t = 60 s, the receiveris not surrounded by buildings and therefore, all the 4 satellites exhibit low probability of fault as seen in Fig. 6(a). However,at t = 74 s, 3 satellites namely, PRN 8, 9 and 22 are blocked by tall buildings and therefore suffer from received signal faults.This is accurately captured using our SLAM-based FDI algorithm as seen in Fig. 6(b).

Figure 7: Receiver localization errors estimated using our SLAM-based FDI are indicated in blue and conventional scalartracking are shown in red. Scalar tracking showed large total positioning errors with RMSE 11.18 m and standard deviationof 33.56 m whereas our SLAM-based FDI, demonstrated low receiver localization errors with RMSE 1.93 m and standarddeviation of 3.97 m.

In Fig. 7, we observed that the conventional scalar tracking, indicated in red, showed large total positioning errors withRMSE 11.18 m and corresponding standard deviation of 33.56 m. However, our SLAM-based FDI algorithm, indicated inblue, demonstrated small receiver localization errors with RMSE 1.93 m and standard deviation of 3.97 m.

4 CONCLUSIONS

We proposed our SLAM-based FDI algorithm which not only detects but also effectively isolates multiple faults caused bybroadcast anomalies and multipath effects. Inspired by SLAM for robotics, we estimate the PVT of GPS satellites in anunknown 3D environment while simultaneously localizing the GPS receiver within it. In particular, we designed a Graph-SLAM framework, where we utilized the sequential GPS pseudoranges, receiver and satellite motion model to optimize thegraph using an M-estimator-based cost function. Later, we assessed the probability of fault associated with each satellite byevaluating its corresponding test statistic against an on-the-fly empirical estimation of its cumulative distribution function.

We validated the performance of our SLAM-based FDI by adding simulated broadcast anomalies to the collected open-skyGPS data. We have shown that our algorithm estimates the probability of fault in a non-faulty satellite with an RMSE of 0.22which indicates high reliability and the 3 faulty satellites as 0.57, 0.81 and 0.61, respectively, which indicates high probability.In the presence of multipath, we demonstrated that our SLAM-based FDI adaptively estimates the multipath-prone satellitesand accurately localizes the receiver with RMSE 1.93 m and standard deviation of 3.97 m.

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