Vertical Protection Level Estimation for DirectPositioning Using a Bayesian Approach

Shubhendra Vikram Singh Chauhan1 and Grace Xingxin Gao2

University of Illinois at Urbana-Champaign1 and Stanford University2

BIOGRAPHIES

Shubhendra Vikram Singh Chauhan received his B.Tech. and M.Tech degree in aerospace engineering from Indian Insti-tute of Technology, Bombay, India in 2016. He received the Institute Silver Medal on graduation. He is currently pursuing hisPh.D. degree at the University of Illinois at Urbana-Champaign. His research interests include robotics, controls and sensorfusion.

Grace Xingxin Gao is an assistant professor in the Department of Aeronautics and Astronautics at Stanford University. Be-fore joining Stanford University, she was an assistant professor at University of Illinois at Urbana-Champaign. She obtainedher Ph.D. degree at Stanford University. Her research is on robust and secure positioning, navigation and timing with applica-tions to manned and unmanned aerial vehicles, robotics and power systems.

ABSTRACT

GPS integrity monitoring is essential to ensure the safety of critical infrastructures, such as power grid, banking, and trans-portation systems. The integrity of a system is assessed using protection levels (PLs). PLs overbound positioning and timingerrors using error models. Positioning errors have multi-modal Gaussian distribution due to changing environmental condi-tions. The number of modes present in the multi-modal distribution is not known and poses a challenge in overbounding er-rors using a multi-modal distribution.Direct positioning (DP) is an unconventional GPS receiver architecture that utilizes signal-in-space and provides position,velocity and time (PVT) solution in a single step. DP is robust against errors that arise due to multipath or signal blockage.In this work, we propose a Bayesian algorithm to estimate PLs for DP. We model the positioning error with a time-varyingGaussian distribution to capture the multi-modal behavior. Our Bayesian algorithm use DP likelihood manifold to estimatePLs. The effect of changing environmental conditions is captured by DP likelihood manifold, which is a function of time-varying variance. In this paper, we focus on overbounding vertical positioning errors of DP using our Bayesian algorithm.We generate 24 hours of stationary GPS data using a high fidelity GPS simulator. We show that the vertical positioning errordistribution for DP is multi-modal. We further validate that our estimated VPL bounds vertical errors for DP.

1 INTRODUCTION

Critical infrastructures, such as power grid, banking, and transportation system use GPS timing and positioning service to en-sure safety [1]. Integrity [2] measures the trustworthiness of a navigation solution. It is one of the most critical requirementsfor safety-of-life applications. Protection levels (PLs) are used to assess integrity requirement for a system [3, 4, 5, 6]. PLsoverbound positioning and timing errors by using error models.Traditional receiver architecture, such scalar tracking loop [7] and vector tracking loop [8], use two steps to provide a position-velocity-time (PVT) solution. Pseudoranges are estimated in the first step and trilateration is performed in the second step toobtain a PVT solution. Pseudorange error models (PEMs) are used to derive PLs. These models assume that each error com-ponent in PEMs is completely characterized by uni-modal symmetric Gaussian distribution. However, the error distribution ismulti-modal due to changing environmental conditions [6]. Overbounding the multi-modal error distribution tails, under suchscheme, may result in underbounding of errors due to asymmetry or bias in the distribution. This may cause loss in integrity[5, 6].

One way to overbound multi-modal error distribution is to use multi-modal Gaussian distribution. In [6], authors show thatthe error distribution is multi-modal. Authors overbound the positioning errors using a bi-modal Gaussian distribution. Usingthis approach, authors improve the overall availability of the system by 50%. This approach implicitly assumes that the num-ber of modes present in the multi-modal error distribution is known. However, the number of modes in the error distribution isdependent on environmental conditions and is unknown.Direct positioning (DP) [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] is an unconventional GPS receiver architecture that directly op-erates in PVT domain. Compared to traditional two steps methods, DP estimates PVT solution in a single step without esti-mating pseudoranges. Thus, DP removes errors arising from convolutions of PEMs.A large amount of literature is available for overbounding positioning errors using PEMs [3, 4, 5, 6], only few are availablefor DP. With best knowledge of authors, there has not been a paper that empirically shows the error distribution for DP. Weexpect the error distribution to be multi-modal due to changing environmental conditions.Prior work [19] on DP-based integrity monitoring discusses Solution-Separation Receiver Autonomous Integrity Monitoringframework. However, this framework is originally designed for PEMs. Another work [20], utilizes the correlation manifoldsgenerated by DP to overbound the positioning errors. This approach is deeply coupled with DP framework. However, authorsuse many empirical parameters to overbound the vertical errors. These parameters are dependent on environmental conditionsand may change with different environmental conditions.Our ultimate goal is to provide PLs for DP in positioning and timing domain. As a starting step, in this work, we focus on es-timating PLs for vertical errors. We also want to verify our hypothesis that vertical errors’ distribution for DP is multi-modal.

1.1 Our Approach and Contributions

Conventional DP integrity monitoring methods use PVT estimate only. In our approach, we use both PVT and noise varianceestimate. We use PVT estimate to generate correlation manifold and we use noise variance estimate to convert correlationmanifold into likelihood manifold. By estimating noise variance, we approximate the vertical positioning errors with time-varying Gaussian distribution. This approximation captures the multi-modality of vertical errors’ distribution. The key contri-butions of our work are as follows

1. We develop a Bayesian algorithm to estimate VPL for DP. Our algorithm is robust to the unknown number of modespresent in the vertical errors’ multi-modal distribution.

2. We validate our algorithm using a high fidelity GPS simulator. We generate 24 hours of stationary GPS dataset. We ob-tain 4 million positioning data points on this dataset. We show that vertical errors for DP have multi-modal distributionand thus verify our initial hypothesis. We further validate that our estimated VPL bounds vertical errors.

The remainder of the paper is organized as follows, a generic DP receiver architecture is presented in section 2. The detailedalgorithm is provided in section 3, which is built on DP receiver architecture. Simulation environment and implementation arepresented in section 4. Results are shown in section 5 and finally, the conclusions of the work are provided in section 6.

2 OVERVIEW OF DIRECT POSITIONING

This section provides an overview of DP receiver. The first subsection describes the mathematical formulation for DP and thesecond subsection provides details for implementation of a generic DP receiver.

2.1 Mathematical Formulation

The objective of DP [9] is to estimate position-velocity-time (PVT) coordinates of a receiver X given the received signal Y ,where:

X =[x y z cδ x y z cδt

]T=

[xx

](1)

(cδt, cδt) denotes receiver specific clock bias and drift multiplied by speed of light. Also, x =[x y z cδt

]Tx =[

x y z cδt]T

. The PVT coordinates are in ECEF coordinate frame. The received signal at time t and at coordinate X,after carrier wipe off is given by

Y (a,X, t) =

M∑i

a(i)g(i)(t− τ i) exp(j2π∆f it

)+ n(t) (2)

where:

• a =[a(1) a(2) ... a(M)

]T ∈ CM are the complex amplitudes of the visible satellites.

• M ∈ N is the number of visible satellites.

• g(i) is the L1 coarse acquisition (C/A) code of the i−th visible satellite.

• τ (i) is the code delay of the i−th visible satellite:

τ (i) =||r(i)||c

+(δt− δt(i)

)(3)

• ∆f (i) is the carrier Doppler shift of the i−th visible satellite:

∆f (i) =−fL1c

{r(i)r(i)

||r(i)||+ c

(δt− δt

(i))}

(4)

• r(i) =[x− x(i) y − y(i) z − z(i)

]Tis the relative vector to the i−th visible satellite.

• (δt(i), δt(i)

) are satellite specific clock bias and clock drift rate.

• n(t) ∈ N (0, σ2) ∈ C is an independent and identically distributed (i.i.d.) Gaussian process, same as the complexadditive Gaussian nose (AWGN).

In DP, receiver’s coordinates are obtained by maximizing the following likelihood

p(y|a,X, σ2) =

(1

πσ2

)Nexp

{−||y −Da||2

σ2

}(5)

where:

• y =[Y (a,X, t1) Y (a,X, t2) ... Y (a,X, tN )

]T ∈ CN is a signal snapshot obtained over t = {tn}Nn=1.

• D(X, t) ∈ CN×M is a matrix containing signal replicas of visible satellites for a given X and t.

• σ2 ∈ R is the noise level of the receiver.

• ||b|| denotes L2 norm of a generic vector b.

Under nice properties of D [18] and using the orthogonality principle, the likelihood [9] is simplified as

p(y|X, σ2) =

(1

πσ2

)Nexp

{−||y||2 − 1

N ||D∗y||2

σ2

}(6)

The maximal likelihood (ML) estimation is then obtained by

XML ≈ argmaxX

1

Ny∗DD∗y = argmax

XR(X, t) (7)

where:

• D∗,y∗ are conjugate transpose of D and y respectively.

• R(X, t) denotes the correlation manifold obtained at coordinate X and time samples t.

2.2 Implementation of Direct Positioning Receiver

The philosophy that time delay and Doppler shift are implicit functions of the receiver’s state, is used to estimate the receiver’sPVT coordinates. These implicit relations are shown in equations 3 and 4 respectively. DP receiver maximizes the correlationmanifold to obtain the PVT coordinates. The implementation of generic DP [21, 16, 17, 14] receiver is described below:

1. In the first step, the DP receiver generates a grid of candidates Xm. Each candidate represents a potential navigationsolution and corresponds to a unique time delay and Doppler frequency shift. The grid is initialized in such a way toensure X remains within the range of candidates.

2. DP generates an expected signal reception, Ym, for each candidate based on the PVT coordinates of the candidate.

Ym(Xm, t) =

M∑i=1

g(i)(t− τ (i))

exp{j2π(fL1t+ ∆f (i)t+ φ(i))

} (8)

where fL1 is L1 carrier frequency (1575.42 MHz) and φ(i) is the carrier phase of the i−th visible satellite. The signalsynchronization parameters are derived from receiver coordinates Xm using the equations 3 and 4. Expected time sam-pled signal is given by time sampling of Ym and is given below

ym =[Ym(Xm, t1) Ym(Xm, t2) ... Ym(Xm, tN )

]T(9)

3. For each candidate, the receiver computes the cross-correlation between the expected reception ym and the receivedsignal y

R(Xm, t) = corr(y, ym) (10)

Correlation manifold is obtained by collectively obtaining correlation values over the grid. This manifold is typicallyunimodal, where the peak is at the candidate closest to the navigation solution. For illustration purpose, a typical corre-lation manifold is shown in Figure 1.

4. The navigation solution is obtained by selecting the candidate that has the highest correlation value

XDP = argmaxm

R(Xm, t) (11)

where XDP denotes the estimated navigation solution provided by DP. This estimate is utilized in the next time step forpopulating the grid candidates, that is performed in step 1.

Figure 1: 2D example [20] showing the correlation manifoldR on the local East-Up plane. The best match denotes the candi-date closest to the receiver.

The high dimensional search space for X is decoupled into two subspaces (1) position and clock bias, x and (2) velocity andclock drift, x. The decoupling is similar to Space Alternating Generalized Expectation (SAGE) algorithms as discussed in[14, 22, 23]. This decoupling is performed to reduce the computation cost.

3 BAYESIAN APPROACH TO ESTIMATE VPL

The overall architecture of our Bayesian VPL estimation algorithm is shown in Figure 2.

Figure 2: Overall architecture of the proposed algorithm

The received raw signal snapshots, y, are used in DP block to estimate PVT coordinates of the receiver and the noise variancein the received signal. The PVT coordinates of the receiver are utilized in creating additional candidates in the local up direc-tion. Altitude likelihood manifold (ALM) is obtained by calculating the correlation manifold at these candidates. In BayesianVPL Estimation block, we utilize the ALM to obtain VPL. The details of each sub-blocks are provided in the following sub-sections.

3.1 Direct Positioning

The generic DP receiver architecture is explained in section 2. We model the positioning errors with time varying Gaussiandistribution. In addition to estimating PVT coordinates, we estimate the variance of the time varying Gaussian distribution inthe received signal. We perform maximal likelihood estimation on equation 6 to estimate time varying noise variance. Thisestimate is given by [18]

σ2 =||y||2 −R(XDP , t)

N(12)

The navigation solution estimate XDP and the estimate of noise variance σ2 is used in the subsequent block to estimate ALM.

3.2 Computing Altitude Likelihood

DP provides two 4 dimensions correlation manifolds. The first one is for position and clock bias. The second one is for veloc-ity and clock drift. In this work, our primary focus is to obtain VPL for a given PVT estimate. The 4D correlation manifold isconverted into 1D correlation manifold along the local up direction. This step is performed to obtain the ALM. The details ofthe algorithm are provided below:

1. Using the navigation solution obtained from DP receiver, we transform XDP into local East-North-UP (ENU) frameusing a reference coordinate. After this we generate, candidates zj in local up direction around the reference coordinate.

2. Signal replicas are generated for these candidates. Note that for these replicas, PVT coordinates are same as XDP ex-cept for the local z coordinate.

yzj =[Yzj (zj , t1) Yzj (zj , t2) ... Yzj (zj , tN )

]T(13)

where yzj denotes the time sampled expected signal at local coordinate zj .

3. In the next step, we perform cross-correlation between the received signal and the expected signal to obtain altitudecorrelation manifold.

R(zj , t) = corr(y, yzj ) (14)

4. We use altitude correlation manifold along with the estimate of noise variance to compute ALM.

p(y|zj) =

(1

πσ2

)Nexp

{−||y||2 − 1

NR(zj , t)

σ2

}(15)

ALM is obtained by calculating the likelihood for each candidate generated in step 1. For illustration purpose, a sample 2Dcorrelation manifold and 1D correlation manifold in local up direction are shown in Figures 3 and 4 respectively. Once thelikelihood is estimated, we proceed to estimate the posterior probability in the next block.

Figure 3: Sample 2D correlation for the candidates in local East and Up direction

-3 -2 -1 0 1 2 30

2

4

6

8

10

12

14

Figure 4: Sample 1D correlation for the candidates in local Up direction

3.3 Bayesian Vertical Protection Level Estimation

The posterior probability is required to estimate VPL. Thus, we use Bayes theorem to estimate the posterior probability andlater use this probability to estimate VPL.

3.3.1 Estimation of posterior probability

Using Bayes theorem the posterior probability is given by

p(z|y) =p(y|z)p(z)p(y)

(16)

In this work, we assume the prior probability, p(z), to be uniformly distributed over the generated local zj candidates. Thenormalizing probability, p(y), is obtained by using the law of total probability

p(y) =

∫p(y|z)p(z)dz (17)

The expression of p(y|z) contains highly non-linear terms. The closed-form solution of the last equation is not available. Wenumerically integrate last equation using the Euler method, over the set of candidates, to obtain p(y). The posterior probabil-

ity under such scheme of integration is then given by

p(z|y) =exp

{R(z,t)σ2N

}∑j

(exp

{R(zj ,t)σ2N

})∆zj

(18)

where:

• R(z, t) denotes altitude correlation manifold at z and time samples t.

• ∆zj denotes the candidates’ separation.

The summation in the denominator is taken over all the candidates zj , generated in the local up direction. The log-sum-exptrick is used to avoid numerical overflow errors. This trick is widely used by deep learning researchers for training the neuralnetwork [24]. The trick is explained as follows:

log

(∑k

exp(bk)

)= log (exp(bmax)) + log

(∑k

exp(bk − bmax)

)(19)

where bk is the kth element of some vector b and bmax = maxk(b) i.e. the maximum element of b. This trick is used toavoid numerical overflow errors that may occur in numerical computation.

3.3.2 Estimation of VPL

For a given integrity risk probability ε, VPL is defined by [6]

p(|z − z| > V PL|y) < ε (20)

The task is to estimate z and an interval I such that for a given ε, the above equation is satisfied. The above equation is simpli-fied to [6]

∫z∈I

p(z|y)dz ≤ ε (21)

There is no unique way to find the interval I . For simplicity, we assume that the probability of being on each side of the inter-val is ε/2. Then the above inequality is numerically solved to obtain the interval I . Once the interval is obtained, upper andlower bound are computed for a given estimate and thus VPL. This algorithm is illustrated in the following figure:

Figure 5: VPL estimation illustration using posterior probability

In the above figure zmin, zmax denotes the minimum and maximum value of the generated local candidates. We incrementallyintegrate the posterior probability from both ends of the distribution. At each incremental integration, we check if the inte-grated probability exceeds ε/2. The first local zj candidate for which condition specified by equation 21 is not satisfied, cor-responds to upper and lower bounds. In Figure 5, these extremes are denoted by zlow and zhigh The estimated VPL is givenby

V PL = |zhigh − zlow| (22)

where || denotes absolute value. Under similar Euler numerical integration scheme, the integration of posterior probability iswritten as

∫z∈I

p(z|y)dz ≈

∑i

(exp

{R(zl,t)σ2N

})∆zl∑

j

(exp

{R(zj ,t)σ2N

})∆zj

(23)

where l is such that zl ∈ I , ∆zl denotes the candidates’ separation and j is such that it covers all the generated candidates, i.e.the sum at the denominator is taken over for all the candidates. Here also, we use log-sum-exp trick to avoid overflow errors.

4 SIMULATION ENVIRONMENT

Real-life outdoor experiments involve uncontrolled conditions. These conditions often are undetectable and difficult to ana-lyze. Obtaining ground-truth in such situations becomes a challenging task. In order to avoid uncontrolled situations and theproblem of obtaining ground truth, we decide to work with simulated dataset. We use a high fidelity GPS simulator to gener-ate GPS raw signals. The simulator provides the functionality of creating custom motion trajectories with adjustable individ-ual satellite power levels and transmitting it directly to the receiver.

Figure 6: High fidelity GPS simulator

The simulator is capable of simulating up to 12 GPS satellites. The number of GPS satellites chosen depends on the almanacand ephemeris files, GPS time, and receiver location that a user specifies. The simulator validates several satellite parametersbefore simulating the satellites. If the satellite parameters are valid at the specified GPS time, the simulator selects the satelliteand uses it to simulate a GPS signal. The simulator continuously updates the Doppler shifts between the satellites and simu-lated receiver so that the simulated signal is as close to a real location as possible.We generate 24 hours of stationary GPS data using this simulator. At each time step, we use 20 × 10−3 seconds of data to getPVT coordinates using DP. The sampling frequency for the generated data is 2.5 MHz, providing us with four million posi-tioning data points. These positioning data points are used to obtain the distribution of the vertical positioning errors.

5 IMPLEMENTATION AND EXPERIMENTAL RESULTS

We implement our derived algorithm on pyGNSS, Python-based software-defined radio (SDR) research suite [21, 25]. pyGNSSprovides the flexibility to test and verify new GPS receiver algorithms. Raw GPS signal samples, generated by NI GNSS sim-ulator, are the input to our pyGNSS. The table 1 tabulates the parameters used in implementation for grid generation.Additional candidates, that are generated to obtain ALM, have a spacing of 1 cm if they are within 5 m of the estimated posi-tion. Otherwise, they have the spacing of 3.5 m.

5.1 Validating Multi-modal Distribution

We use GPS synthetic data and test it with DP to obtain vertical positioning errors distribution. We perform DP, as explainedin section 2, on the generated dataset. The estimated receiver coordinates are compared with ground truth to obtain vertical

Domain Axis Span Spacing DimPosition East,North,Up [−351.6, 351.6] (m) 46.88 (m) 15Velocity East,North,Up [−17.58, 17.59] (m/s) 2.34 (m/s) 15

Time cδt [−351.6, 351.6] (m) 46.88 (m) 15

Time Drift cδt [−0.22, 0.22] (m/s) 0.03 (m/s) 15

Table 1: Grid samples for implemented DP

errors. The histogram plot of the vertical positioning errors is shown in Figure 7.

Figure 7: Histogram plot of vertical positioning errors shows multi-modal distribution

-6 -4 -2 0 2 4 6

Standard Normal Quantiles

-10

-8

-6

-4

-2

0

2

4

6

8

Qu

an

tile

s o

f In

pu

t S

am

ple

QQ plot of true vertical error versus standard normal

True Vertical Error

Figure 8: Vertical positioning error distribution is non-Gaussian as displayed in QQ plot

In Figure 8, a quantile-quantile (QQ) plot is shown. The x axis of the plot denotes quantiles taken from a standard normal dis-tribution and y axis of the plot denotes quantiles taken from vertical positioning error distribution. The deviation of verticalpositioning error distribution from red dotted lines shows that the distribution is non-Gaussian. This is also verified in Fig-ure 7, in which the multi-modal distribution behavior of vertical positioning errors is visible. There are roughly four milliondata points present in Figures 7 and 8. Overbounding of multi-modal distribution using a unimodal Gaussian distribution mayresult in reduced performance or loss in integrity for a navigation system.

5.2 Estimated VPL

The proposed algorithm explained in section 3 is implemented and tested on the simulated dataset. The estimated VPL isshown in Figure 9. The subplot in Figure 9, is a zoomed-in version of the top plot. To compare the distributions of estimatedVPL and vertical positioning errors, two QQ plots are generated and are shown in Figure 10 and 11.

Figure 9: Vertical positioning errors and estimated VPL

-6 -4 -2 0 2 4 6

Standard Normal Quantiles

-4

-2

0

2

4

6

8

10

12

14

Qu

an

tile

s o

f In

pu

t S

am

ple

QQ Plot of Sample Data versus Standard Normal

Absolute vertical error

VPL

Figure 10: QQ plot of vertical positioning errors and estimated VPL vs standard normal

Figure 11, shows that the distributions of estimated VPL and vertical positioning errors are similar for errors smaller than 6m.Therefore, the derived algorithm is able to capture the multi-modal behavior. Even for larger error, the estimated VPL over-bounds the vertical errors. This is evident from the histogram plot shown in Figure 12.

6 CONCLUSIONS

We proposed a Bayesian algorithm to estimate VPL for DP. Our approach is robust to multi-modal distribution of positioningerrors. We validated our algorithm using a high fidelity GPS simulator. We generated 24 hours of stationary GPS data and

0 1 2 3 4 5 6 7 8 9 10

Absolute vertical error quantiles

0

2

4

6

8

10

12

14

16

18

VP

L q

ua

ntile

s

QQ plot of vertical error data versus VPL data

Figure 11: QQ plot of vertical positioning errors vs estimated VPL shows that vertical errors and estimated VPL have similardistribution for small vertical errors.

Figure 12: Histogram plot of vertical positioning errors and estimated VPL shows that estimated VPL bounds the verticalerrors.

obtained 4 million positioning data points. We showed that vertical positioning errors for DP has multi-modal distribution. Wevalidated that our estimated VPL bounds vertical positioning errors.

ACKNOWLEDGEMENT

This material is based upon work supported by the Department of Energy under Award Number DE-OE0000780.This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the UnitedStates Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumesany legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific com-mercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute orimply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views andopinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agencythereof.

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