robust toa-based navigation under measurement model

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Robust TOA-based Navigation under MeasurementModel Mismatch in Harsh Propagation Environments

Gaël Pagès, Jordi Vilà-Valls

To cite this version:Gaël Pagès, Jordi Vilà-Valls. Robust TOA-based Navigation under Measurement Model Mis-match in Harsh Propagation Environments. IEEE 23rd International Conference on Intelli-gent Transportation Systems (ITSC), Sep 2020, Rhodes (virtual conference), Greece. pp.1-6,�10.1109/ITSC45102.2020.9294221�. �hal-03274794�

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https://doi.org/10.1109/ITSC45102.2020.9294221

Pagès, Gaël and Vilà-Valls, Jordi Robust TOA-based Navigation under Measurement Model Mismatch in Harsh

Propagation Environments. (2020) In: IEEE 23rd International Conference on Intelligent Transportation Systems

(ITSC), 20 September 2020 - 23 September 2020 (Rhodes, Greece).

Robust TOA-based Navigation under Measurement Model Mismatch inHarsh Propagation Environments

Gaël Pagès and Jordi Vilà-Valls

Abstract— Global Navigation Satellite Systems (GNSS) is thepositioning technology of choice outdoors but it has many limi-tations to be used in safety-critical applications such IntelligentTransportation Systems (ITS). Namely, its performance clearlydegrades in harsh propagation conditions, these systems arenot reliable due to possible attacks, may not be available inGNSS-denied environments, and using standard architecturesdo not provide the precision needed in ITS. Among the differentalternatives, Ultra-WideBand (UWB) ranging is a promisingsolution to achieve high positioning accuracy. The key pointsimpacting any time-of-arrival (TOA) based navigation systemare i) transmitters’ geometry, and ii) a perfectly known trans-mitters’ position. In this contribution we further analyze theperformance loss of TOA-based navigation systems in real-lifeapplications where we may have both transmitters’ positionmismatch and harsh propagation conditions, i.e., measurementscorrupted by outliers. In addition, we propose a new robustfiltering method able to cope with both effects. Illustrativesimulation results are provided to support the discussionand show the performance improvement brought by the newmethodology with respect to the state-of-the-art.

I. INTRODUCTION

Reliable and precise position, navigation and timing (PNT)information is fundamental in safety-critical applicationssuch as Intelligent Transportation Systems (ITS), automatedaircraft landing or autonomous unmanned ground/air ve-hicles (robots/drones), to name a few. In addition, in thecontext of ITS, this is not only of capital importance forthe autonomously navigating system but also for vulnerableroad users such as cyclists and pedestrians, and systemscollaterally using this information, i.e., in traffic controland for emergency services. Even if the main source ofpositioning information are still Global Navigation SatelliteSystems (GNSS), they lack of reliability and accuracy inconstrained environments such as highly populated areas,which limit their adoption as standalone PNT system. Forinstance, GNSS may be affected by attacks such as jammingand spoofing [1], or be severely affected in harsh propagationconditions [2]. Moreover, standard GNSS may not providethe precision needed in the ITS context, i.e., sub-meterlane-level. Several GNSS carrier phase-based precise PNTsolutions exist to improve the latter, for instance, Real-TimeKinematics (RTK) [3, Ch. 26] or Precise Point Positioning(PPP) techniques [3, Ch. 25], but they need either a referencestation or real-time precise corrections, and are even more af-fected by harsh propagation conditions than standard GNSS

This work has been supported by the DGA/AID projects2018.60.0072.00.470.75.01 and 2019.65.0068.00.470.75.01.

Authors are with ISAE-SUPAERO, University of Toulouse, Toulouse,France. {name.surname}@isae-supaero.fr

code-based techniques. In addition, these systems may not beavailable at all in the so-called GNSS-denied environments.

Several alternatives to GNSS exist, ranging from theexploitation of cellular signals (LTE/5G) or other signals-of-opportunity, the combination with local inertial navigationsystems (INS) or the use of vehicle-to-everything (V2X)communications to obtain peer-to-peer measurements. Adifferent approach is to consider dedicated infrastructure,specifically designed to provide precise ranging measure-ments. This is the case of Ultra-WideBand (UWB) tech-nologies, which is exploited in this contribution. They aretypically based on time-of-arrival (TOA) two-way rangingmeasurements and can achieve a sub-decimeter level rangingaccuracy in line-of-sight (LOS) nominal conditions [4]–[8].With respect to other ranging technologies, UWB has theadditional advantage to be more robust to interferences.

In general, UWB-based navigation has the fundamentaldrawback that anchor nodes (transmitters) position is as-sumed to be perfectly known, which may not be the casein real-life applications. Given the sub-decimeter nominalaccuracy of the system, this may have a strong impact on thefinal performance as it has been recently shown in [9]. This isalso a critical point if UWB ranging measurements are usedin multi-sensor data fusion platforms, for instance in com-bination with GNSS, which require a common navigationcoordinate frame [10]. Therefore, in real-life applications, amismatch on the transmitters’ position is a key point to becarefully analyzed for reliable UWB-based navigation.

In this contribution, based on the preliminary results in [9]where we proposed an augmented state extended Kalmanfilter (EKF) to cope with possible anchor position mis-match under nominal Gaussian conditions, we further exploreUWB-based navigation in realistic scenarios. We considerthat several transmitter to receiver links may be affectedby multipath or non-line-of-sight (NLOS) conditions, thenassuming the typical contamination model arising in robuststatistics [11] for certain ranging measurements. Withoutmodel mismatch, the standard solution is to consider a robustregression-based EKF, but this methodology does not applywhen both outliers and model mismatch are present, mainlybecause the filter is not able to distinguish between truemeasurement outliers and measurements which deviate fromthe nominal due to the mismatch. Instead, we propose touse a robust weighting function as uncertainty indicatorwithin the augmented state EKF, which is used to adapt themeasurement noise covariance matrix. Illustrative simulationresults are provided to support the discussion ans show theperformance improvement brought by the new methodology.

Authorized licensed use limited to: ISAE. Downloaded on June 17,2021 at 07:39:43 UTC from IEEE Xplore. Restrictions apply.

II. PROBLEM FORMULATION

Consider the positioning of a mobile agent, where at timet both position and velocity are to be inferred, pm,t =[xm,t, ym,t, zm,t]

T and vm,t = [vx,t, vy,t, vz,t]T . If L trans-

mitters (Tx) (i.e., anchor nodes or cellular base stations),within the communication range of the target, are located atfixed positions pi = [xi, yi, zi]

T , with i = 1, . . . , L, then themeasured Tx to agent distance is given by :

zi,t = ||pm,t − pi||+ ni,t, (1)

where || · || is the L2-norm,

di,t = ||pm,t − pi||

=√

(xm,t − xi)2 + (ym,t − yi)2 + (zm,t − zi)2,

and ni a measurement noise. The full set of availableobservations (measurement equation) is written as: z1,t

...zL,t

︸︷︷︸

zt

=

||pm,t − p1||...

||pm,t − pL||

︸︷︷︸

ht(xt)

+

n1,t...

nL,t

︸︷︷︸

nt

. (2)

In real-life application the number of available observa-tions may be time-varying, then the system must be awarethat the size of the measurement vector may be changing overtime. Regarding the dynamics of the agent, for simplicity wemay consider a constant velocity model (process equation),[

pm,tvm,t

]︸︷︷︸

xt

=

[I3 ∆tI303 I3

]︸︷︷︸

F

[pm,t−1vm,t−1

]︸︷︷︸

xt−1

+

[03

wv,t−1

]︸︷︷︸

wt−1

,

(3)with wt−1 ∼ N (0,Q). Both (2) and (3) define the state-space model (SSM) formulation of the problem. To furtherintroduce the main problem to be considered in this contribu-tion, we define the following nominal and mismatched SSMs,and how to account for non-nominal propagation conditions.• Nominal SSM: in the nominal case (i.e., what the mobile

agent would like) pi are the true Tx positions and themeasurement noise is zero-mean Gaussian with equalvariance (all measurements have the same quality), nt ∼N (0L,R), R = σ2

r,iIL.• Mismatched SSM: in real-life applications we may have

a partial knowledge or uncertainty in a subset of Txpositions. In this case, the position mismatch on a subsetUe of Le ≤ L Tx is written as

pi = pi + ∆pi for i ∈ Ue, (4)

with ∆pi = [∆xi ∆yi ∆zi]T the position error on the

i-th Tx, which can be viewed as a bias in its position. Themismatched measured distance (for i ∈ Ue) is

zi,t = ||pm,t − pi||+ ni,t

= ||pm,t − (pi + ∆pi)||+ ni,t

=√d2i,t + εt(pm,t,∆pi) + ni,t, (5)

with εt(pm,t,∆pi) = −2(pm,t − pi)T∆pi + ∆pTi ∆pi,

and pm,t and ∆pi (or equivalently pi) being unknown.We have in this case that zt = ht(xt) + nt, with the newmeasurements zTt = [z1,t, . . . , zLe,t, zLe+1,t, . . . , zL,t] andthe corresponding measurement equation

ht(xt) =

||pm,t − p1||...

||pm,t − pLe||

||pm,t − pLe+1||...

||pm,t − pL||

, (6)

where xt can be the original one in (3) or an augmentedstate (see next Section III-B).

• Harsh propagation conditions: it is known that, for in-stance in urban environments or indoors, there are severalpropagation effects which deviate from Gaussianity, thenbeing crucial to design robust methods in order to obtaina reliable solutions. A possible way to account for theseconditions is to consider non-Gaussian measurement noisedistributions, i.e., Student t, Laplace or skew t [12], [13].Another approach is to consider the typical contaminationmodel arising in robust statistics [2], [11]: consider aproportion 1− ε of observations under nominal Gaussiannoise, and another proportion 0 ≤ ε ≤ 1 of observationscontaminated by an unknown distribution,

ni,t ∼ (1− ε)N (0, σ2r,i) + εH, (7)

where H is an arbitrary contaminating distribution ac-counting for possible outliers (i.e., corrupted observations),for instance, a non-zero mean Gaussian distribution withσH � σr,i.The main question is: how do we estimate the agent’s

position and velocity under both Tx position mismatch andnon-nominal propagation conditions?

III. REFERENCE EKF-BASED SOLUTIONS

A. Standard Extended KF SolutionConsidering the SSM in (2) and (3), it is easy to design

an EKF [14], which needs a linearized (approximated) mea-surement equation, given by the following Jacobian matrix(evaluated at a given point x0 and with ui,t = ||p0 − pi||),

Ht =∂ht(xt)

∂xt

∣∣∣∣x0

=

x0−x1

u1,t

y0−y1u1,t

z0−z1u1,t

0T3...

......

...x0−xL

uL,t

y0−yLuL,t

z0−zLuL,t

0T3

.(8)

The standard TOA-based navigation is obtained by applyingthe EKF to the assumed model (2)-(3):

xt|t−1 = Fxt−1|t−1

Σx,t|t−1 = FΣx,t−1|t−1F + Q

Kt = Σx,t|t−1HTt (HtΣx,t|t−1H

Tt + R)−1

xt|t = xt|t−1 + Kt(zt − ht(xt|t−1))

Σx,t|t = (I−KtHt)Σx,t|t−1


with Ht = ∂ht(xt)/∂xt|xt=xt|t−1. Notice that the linear

KF recursion is only valid (t ≥ 1) if Σx,0|0 = Cx0and

x0|0 = E[x0] (mean and covariance of the initial state) [15],which in practice are unknown, then an important point isthe filter initialization. A practical solution is to use as initialestimate the linear minimum variance distortionless response(LMVDR) estimator, which coincides with the weighted leastsquares estimator (WLSE) of x1, and is given by

x1|1 = Σx,1|1HT1 R−1z1, Σx,1|1 =

(HT

1 R−1H1

)−1, (9)

which in this case, due to the nonlinearity of the measure-ment equation must be replaced by an iterative WLSE.

In case of model mismatch, this EKF-based solution isblind because it directly uses the assumed (mismatched)measurements di,t = ||pm,t − pi||. This can always be usedand provides an acceptable (i.e., depending on the applicationrequirements) solution if d2i,t >> |εt(pm,t,∆pi)|.

B. Augmented State EKF Solution

An alternative to the previous standard EKF which doesnot take into account the possible model mismatch was re-cently proposed in [9]. The idea is to consider an augmentedstate, xt = [pm,t vm,t p1,t . . . pLe,t]

T , that is, to includethe (partially) unknown pi into the state to be estimated. Theprocess equation in this case is given by

pm,tvm,tp1,t

...pLe,t

︸︷︷︸

xt

=

I3 ∆tI3I3

I3Le

︸︷︷︸

F

xt−1 +

03

wv,t−103

...03

︸︷︷︸

wt−1

,

(10)

and the measurement equation as in (2). The correspondingJacobian matrix (dimension L× (6 + 3Le)) is

Ht

∣∣∣xt=x0

=

l1,t 0T3 −l1,t...

.... . .

lLe,t 0T3 −lLe,t

lLe+1,t 0T3 0T3 · · · 0T3...

......

. . ....

lL,t 0T3 0T3 · · · 0T3

,

(11)with li,t =

[x−xi

||p−pi||y−yi||p−pi||

z−zi||p−pi||

]the LOS pointing

vector. It has been shown in [9] that this provides muchbetter results than the standard EKF if not all the Tx positionsare partially unknown (i.e., Le < L). In order to take intoaccount the Tx position uncertainty, the measurement noisevariance of the possible mismatched measurements is set to:

[R]i,i = σ2r,i + σ2

p,i. (12)

A possible choice of σ2p,i is to consider a maximum bias,

for instance, an uniformly distributed random position bias∆pi ∈ [−bi,bi], with bTi = [bx,i by,i bz,i]. If the bias isequal on every coordinate, bx,i = by,i = bz,i = bi, thenσ2p,i = b2i /3.

IV. ROBUST EXTENDED KALMAN FILTER

How to deal with measurements under the contaminationmodel in (7) has been studied for several decades in thefield of robust statistics. A recent publication provides anexcellent introduction with several applications to practicalsignal processing problems [11]. In the sequel we give firsta brief introduction on the idea behind the most basic robustmethods, and then how a robust weighting function can beused to robustify the augmented state EKF of interest.

A. The Idea behind Robust Estimation

We consider as an example the linear regression problem,y = Hx + n, where we want to estimate x and the noisecomponents are i.i.d. If we define the residuals r = y−Hxthe solution is given by the least squares (LS) estimator,

xLS = arg minx||y−Hx||2 ⇒ arg min

x

n∑i=1

(ri(x)

σ

)2

(13)

where we introduced the normalization of the residuals.Notice that a single outlier can destroy our estimate, then notbeing a robust solution. Instead of using a quadratic functionof the residuals, we can consider a general loss function ρ (·)(and its derivative for the minimization ψ(x) = ∂ρ(x)

∂x ) as

x = arg minx

n∑i=1

ρ

(ri(x)

σ

), (14)

n∑i=1

ψ

(ri(x)

σ

)∂ (ri(x)/σ)

∂x= 0, (15)

and for instance, ρLS (x) = x2 and ρ`1 (x) = |x| correspondto the LS and `1 estimators. The most common ρ (·) is theso-called Huber function

ρ(x) =

{x2 if |x| ≤ a2a|x| − a2 if |x| > a

, (16a)

ψ(x) =

{x if |x| ≤ aa sign(x) if |x| > a

, (16b)

w(x) =

{ψ(x)/x, if x 6= 0ψ′(0), if x = 0

= min

{1,

a

|x|

}, (16c)

where the value a is chosen to obtain a given efficiency(deviation from the optimal under nominal conditions), forinstance, a = 1.345 for a 95% efficiency. The idea is thatresiduals with large errors are downweighted. To solve theproblem in (15) we need the residuals standard deviation σ,or a robust estimate of it, for instance the normalized medianabsolute deviation (MAD), σM , defined as

σM (x) = cm Med(|x−Med(x)|), (17)

where Med(x) is the median and cm a normalizing constant(typically cm = 1.4815). Considering the weight function in(16c) it is easy to see that (15) can be rewritten as

n∑i=1

w (ri/σM )riσM

∂ (ri/σM )

∂x= 0, (18)

which can be solved by an iterative reweighted LS and isthe so-called regression M-estimator.


B. Robust Regression EKF

The robust regression M-estimator introduced in the previ-ous section IV-A can be used within the EKF framework inorder to obtain a robust EKF [11]. Notice that the problemof interest is the one where we may have outliers in theobservations zt and the residuals are related to the innovationsequence zt−ht(xt|t−1). A first approach is to use a robustscore function ψ(·) in order to directly downweight theinnovation vector, then not modifying the EKF recursion.Another more general approach is to reformulate the EKFas a regression problem and then use at every t a robustregression M-estimate of xt. Refer to [11, Chap. 7] fordetails. Notice that both approaches were not developedto cope with a possible model mismatch and they are notdirectly suited for the navigation problem in this contribution.The underlying problem to apply these techniques to themismatched SSM is that the filter is not be able to distinguishbetween true measurement outliers and measurements whichdeviate from the nominal due to the mismatch.

C. Robust Weighting Uncertainty Indicator for Robust EKF

An alternative to the robust regression EKF is to use therobust weighting function in order to update the statisticalcharacteristic of the measurement noise variance. Indeed,the performance of the EKF is highly affected by themeasurement covariance matrix, R, therefore one or morecontaminated measurements will have a significant impacton the filter performance, i.e., estimated state.

The proposed method is based on the values taken bythe normalized residuals (i.e., innovations) as introducedin Section IV-A. In order to reduce the impact of thecontaminated measurements on the estimator, the idea isto increase the variance of each contaminated measurementdepending on the value of |ri/σM | . To do so, we proposein this study to use the square inverse of Huber’s weightingfunction, as shown in Figure 1, to adjust the measurementnoise variances. Thus, in the case of large residuals, i.e.|ri/σM | > a, and in the framework of anchor positionmismatch, the variance can be formulated as,

[R]∗i,i =1

(w (ri/σM ))2 [R]i,i. (19)

By replacing w (ri/σM ) by expression (16c), the latterequation can be rewritten as

[R]∗i,i = r2i[R]i,ia2σ2

M

. (20)

According to this last equation, we note that the varianceassociated with the contaminated measurement is propor-tional to squared residual and that it is able to cope withthe anchor position mismatch.

V. ILLUSTRATIVE UWB-BASED NAVIGATION EXAMPLE

The performance of the proposed robust estimator inSection IV-C, named MRCEKF, is assessed in a simulationenvironment and compared to the standard EKF (SEKF) inSection III-A, the mismatch EKF (MEKF) in Section III-B,

Fig. 1. Square inverse of Huber’s weighting function.

and the robust regression EKF (RREKF) from [11, Chap. 7].The Huber’s score function (a = 1.345), as defined in sectionIV, is used to compute the RREKF and the MRCEKF. Theresults are obtained from 100 Monte Carlo (MC) runs.

A. Simulation Scenario

We consider L = 8 anchors and a realistic mobileagent trajectory, shown in Figure 2. The position of fiveanchors (A4 - A8) is considered to have a 3-dimensionalbias which is drawn randomly in [−0.5, 0.5] m, for eachMC run. The unknown state vector to be estimated is thenxt = [pm,t,vm,t, p4, . . . , p8]

T .The initial position and velocity of the agent were set

to pm,0|0 = (0,−2, 0.5)T and vm,0|0 = (0, 0, 0)T plus arandom initial bias of ±0.5 m and ±0.01 m.s−1, respec-tively. The noise of the inlier observations is described by azero-mean Gaussian distribution whose variance was set toσ2ri,LOS = 0.01 m2. Similarly, the NLOS effect (outliers)

in the observations is modeled as a zero-mean Gaussiandistribution with a variance α2 times larger than that ofthe inlier observations. The mixed LOS/NLOS conditionsare modeled as a Markovian process that switches betweenthe LOS and NLOS conditions. Different percentages, ε, forthe the probability of being in NLOS conditions, and outliermagnitudes, α, are considered and indicated in Table I. Thenumber of anchors in NLOS conditions is drawn randomlywithin the 8 anchors, and the number of regions and lengthof each region in mixed LOS/NLOS conditions are alsorandomly generated.

TABLE IPARAMETERS FOR THE MONTE CARLO SIMULATION.

Number of runs 100Range variance noise (m2) 0.01Number of anchors 8Number of mismatched anchors Le = 5Anchor position bias (cm) [−50,+50]Outlier percentage ε 10− 25− 50Outlier magnitude α 1− 5− 10− 30− 60Number of NLOS anchors 2− 4− 6− 8

B. Results

First, we show in Figure 2 a single realization, and thecorresponding four estimates, to clearly illustrate both the


Fig. 2. 2D (left) and 3D (right) trajectories and the corresponding estimates. The scenario is based on 5 anchors out of 8 having a random positionbias of ±0.5m (blue circles). The mixed LOS/NLOS conditions are generated for a specific region of the trajectory and concerns all 8 anchors. They aremodeled as a Markovian process with the following parameters: σri,NLOS = 3m, ε = 25%.

impact of the mismatched anchors (blue circles) and harshpropagation conditions (outliers), which are evident in suchresults. The former can be seen in the 3D plot wherethe standard SEKF and RREKF are not able to deal withthe position mismatch and are affected by a misalignmentof the estimated trajectory. Even if the RREKF correctlydiscards outliers (see 2D plot) it trusts the mismatched anchorpositions as being the real ones. The SEKF is not able tocope with neither the mismatch nor outliers. Regarding theaugmented state techniques, the MEKF is able to deal withthe mismatch but not with the outliers. Notice that only thenew MRCEKF provides a smooth trajectory properly dealingwith both position mismatch and corrupted measurements.

Figure 3 shows the average horizontal and vertical rootmean square errors (RMSE) obtained with the four esti-mators as a function of the NLOS standard deviation (i.e.,σri,NLOS = ασri,LOS). We provide the results obtained for4 and 8 anchors under NLOS conditions, and ε = 25 and50%. Notice that other configurations were also tested but weonly illustrate the most relevant results. As expected, for agiven percentage of contamination, the MEKF breaks downrapidly as the amplitude of the outliers increases. Indeed,the MEKF tries to compensate the modeling error due tothe mismatch, then measurements corrupted by outliers areunderstood by the filter as a bias in the anchors’ position.In comparison, the SEKF is less affected by this behavior,with no clear performance breakdown, but in general thismethod provides a worse estimate when compared to the tworobust estimators. On the contrary, the RREKF estimator isrelatively stable, thus correctly dealing with outliers, but itsnominal performance is degraded because of the mismatch.

For low outlier magnitudes, α, it is clear when comparingthe outcome of both MEKF and RREKF that a methodbeing able to cope with both mismatch and outliers shouldretain the best qualities of both estimators. Indeed this is thecase with the new MRCEKF which is relatively stable andcopes perfectly with contaminated observations and anchorposition mismatch, up to a certain level of contamination.This new robust method outperforms the rest when the

number of contaminated measurements is less or equal than6 and ε ≤ 50%. From the results analyzed it can beconcluded that for ε < 50%, α ≤ 3m and the number ofanchors in NLOS conditions ≤ 6, the MRCEKF approachguarantees horizontal and vertical performances below thetens of centimeters, unlike the other approaches.

VI. CONCLUSION

In this contribution we have shown the impact of bothmodel mismatch (i.e., transmitters’ position uncertainty) andharsh propagation conditions (i.e., measurements corruptedby outliers) in realistic UWB-based navigation systems, witha non-negligible performance loss if not properly taken intoconsideration. A possible way to mitigate both effects is: i) tointroduce the uncertain anchors’ position into the state to betracked by a KF-like method, and ii) to resort to robust statis-tics for the design of robust filtering techniques. We proposeda robust filter able to cope with both model mismatch andcorrupted measurements by leveraging a robust weightingfunction to adapt the measurement noise statistics. This newapproach provides a significant performance improvementunder several configurations with respect to the state-of-the-art, being a promising solution to be further explored.

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(b)

Fig. 3. Average horizontal (a) and vertical (b) RMSE results for four relevant examples using the standard EKF (SEKF), the mismatch EKF (MEKF),the robust regression EKF (RREKF) and the mismatch robust weight-based EKF (MRCEKF).

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