Multipath Detection and Mitigation from GNSS Observations Using Antenna Arrays

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Multipath Detection and Mitigation from GNSS Observations Using Antenna Arrays

Item Type Conference Paper

Authors Ahmed, Mohanad; Ballal, Tarig; Liu, Xing; Al-Naffouri, Tareq Y.

Citation Ahmed, M., Ballal, T., Liu, X., & Al-Naffouri, T. Y. (2023).

Multipath Detection and Mitigation from GNSS Observations Using Antenna Arrays. 2023 IEEE/ION Position, Location and Navigation Symposium (PLANS). https://doi.org/10.1109/

plans53410.2023.10139949 Eprint version Post-print

DOI 10.1109/plans53410.2023.10139949

Publisher IEEE

Rights This is an accepted manuscript version of a paper before final publisher editing and formatting. Archived with thanks to IEEE.

Download date 21/06/2023 06:59:34

Link to Item http://hdl.handle.net/10754/692535

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Multipath Detection and Mitigation from GNSS Observations Using Antenna Arrays

Mohanad Ahmed, Tarig Ballal, Xing Liu, and Tareq Y. Al-Naffouri Computer, Electrical and Mathematical Science and Engineering Division

King Abdullah University of Science and Technology (KAUST) Thuwal, Saudi Arabia

{mohanad.ahmed,tarig.ahmed,xing.liu,tareq.alnaffouri}@kaust.edu.sa,

Abstract—We consider the problem of detecting and mitigating the effect of multipath on GNSS observations. In particular, we focus on carrier-phase observations that are collected at an array of GNSS antennas. We exploit a special antenna array geometry, synchronization among the GNSS receivers, and an attitude estimate provided by an attitude filter or an IMU, to develop a technique to identify the satellite observations that are contaminated with multipath. The proposed technique leverages the antenna geometry to rabidly estimate the attitude parameters for various satellite combinations. Next, a dedicated decision-making algorithm is used to identify the satellite observations affected by multipath. Our simulation results demonstrate the effectiveness of the proposed approach in detecting the occurrence of multipath with high success rates. By rejecting the multipath-affected observations, we show remarkable performance gains when attitude determination is considered as an example application.

Index Terms—GNSS Multipath, Multipath Mitigation, Atti- tude, NLOS, Antenna array,

I. INTRODUCTION

Global Navigation Satellite Systems (GNSS) applications in the modern world are almost innumerable. They have broad applications in civilian navigation/localization and are an enabling technology for many applications. Autonomous vehicles and other high precision demanding/safety critical applications motivate research in minimizing the sources of error. The main source of error in GNSS in urban environments is multipath signals [1], [2]. There are several sources that contribute to GNSS observation error. However, the impact of many these errors can be effectively mitigated. For example, ionospheric and tropospheric, as well as clock, errors are greatly mitigated by the (single and double) differencing of the observations. Despite the resounding success of GNSS in many applications, remaining threats are still around and has to be dealt with. In particular, various GNSS applications are extremely sensitive to jamming, spoofing and multipath effects. It is widely perceived that jamming and spoofing are the two most dangerous threats to the operation of systems that rely on GNSS to deliver sensitive information, such as position and timing. Efforts to design GNSS receivers that are less vulnerable are ongoing in what can be regarded as the most critical mission for the GNSS research community.

While jamming and spoofing can render a GNSS-dependent application completely useless, the multipath effect, on the other hand, can degrade the performance of a GNSS application substantially. The presence of severe multipath can

result in huge errors that make meeting the application’s needs impossible. The multipath problem is usually associated with urban environments and urban canyons where the high building density and large building sizes exacerbate the issue.

The absence of a systematic to deal with multipath, makes it one of the urgent issues for the research community to work on. This is further motivated by the very immediate and imminent need for precise and reliable navigation in urban environments required to enable autonomous mobility.

The difficulty in dealing with multipath is typically at- tributed to its coherence with the LOS signal [3], [4]. At relatively low receiver speeds, multipath signals are still in the tracking bandwidth of the locked loops [1]. Urban reflectors are typically within tens of meters from the receiver and hence fit within a chip period.

Numerous methods have been developed in the literature to detect or mitigate multipath effects at various levels in the GNSS signal path. At the antenna level, the existing techniques include integrated multipath limiting array, ground- based augmentation system, RF absorbent ground planes, and polarization diversity methods [2], [5], [6]. At the signal level, correlation signal processing techniques like signal gat- ing, statistical model parameter estimation techniques, and synthetic aperture techniques have been addressed [7]–[10].

At the observation level, there are carrier code smoothing techniques, dual right- and left-hand polarization measurement tracking with SNR weighing of measurements, and consistency checking-based methods through residuals or other metrics [11]–[13]. Coupled with consistency checking, innovation filtering can be performed in the navigation filter [14]. Finally, more techniques are available at the post-processing level with no real-time requirements [1], [15].

Methods making use of antenna arrays have been proposed in the literature. Many of these methods are beamforming methods operating at the signal level, for example [16]–[22].

The common factor among these methods is that the antenna array elements are separated by up to half-wavelength. The close separation between antennas (<10cm for L1 signals) leads to cross-coupling issues between the antennas [22].

Designing the beamformer to very selectively null out the multipath while not decreasing the main signal is a challenging task. To achieve high selectivity, larger baselines hence higher angular resolution (and more antenna elements) are required.

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Fig. 1. A vehicle in an Urban canyon with different reception scenarios

The half wavelength condition between array elements en- sures that the observed phase at the antennas is unambiguous.

Alternatively, the ambiguity can be tolerated in the phase observations and corrected at a later stage. The process of estimating the unknown integers is known as phase disam- biguation. This idea was proposed in a very low computational complexity form in [23], [24] and utilized in GNSS context for attitude determination in [25]–[28]. Theoretical generalization of this idea under the name of2q-order difference arrays was proposed in [29].

In this paper, we propose using a synchronized receiver array with five elements. Building on the ideas in [23]–[25], multipath mitigation is accomplished by utilizing two metrics related to the direction of arrival of the signal and the attitude estimate of the receiver array. The proposed method we believe offers the following advantages:

1) Availability of attitude estimates starting at three visible satellites.

2) Relaxation of antenna array elements proximity requirements and hence reduction in cross-talk.

3) Lower computational costs as compared to other ambiguity resolution methods.

4) Single Epoch as compared to C/N0 or other time series processing methods.

5) Higher angular resolution due to increased baseline.

These advantages come with the burdens of requiring at least five receiver elements and synchronized processing of the receiver element signals. The effective SNR of the phase observations is also reduced by the square of the increase in the baseline. The tradeoff between these advantages and costs is, of course, an engineering exercise.

The rest of this paper is organized as follows: Section II presents the problem formulation. The proposed mitigation algorithms are presented in Section III. The proposed algorithm is verified using simulated GNSS carrier phase data in SectionIV. Finally, concluding remarks and future works are given in SectionV.

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Fig. 2. Receiver array configuration (a) signal specified by amplitudea^s_i, initial phaseθ^s_i and incoming directionq^s_i impinging on a receiver array.

(b) Top view of the antenna configuration showing inter-element spacing and baseline direction vectorsbx,by.

II. SYSTEMSETUP ANDPROBLEMFORMULATION

Consider the scenario depicted in Fig. 1. A vehicle is moving in an urban area. Depending on the specific geometry of the surrounding, signals from satellites above the horizon arrive in different combinations at the receiver antennas, either through a direct path or an indirect path (i.e., multipath). We make the assumption that the antenna array is reasonably small (i.e., the elements are within a few wavelengths from one another) such that all elements experience the same signal paths. This is a reasonable assumption for signal reflectors that are not too close to the receivers since GNSS satellites are very far away relative to the inter-antenna distances. From a specific satellite, zero or one direct path signal can be received.

We can also have zero or more indirect paths. Let the direct path amplitude be a^s₀ and the indirect paths have amplitudes {a^s_i}^N_i=1^{M P}^s , where N_{M P}^s is the number of indirect paths from satellite s. We have four different reception scenarios:

a) Line of Sight (LOS) Scenario: One direct path, no indirect paths(a^s₀≫0, a^s_i ≈0for alli >0).

b) Non Line of Sight (NLOS) Scenario: No direct paths, but some indirect paths.(a^s₀≈0, a^s_i ≫0 for somei >0) c) Multipath Scenario: direct and indirect paths. (a^s₀ ≫

0, a^s_i ≫0for some i >0)

d) Blocked Scenario: no direct or indirect paths. (a^s₀ ≈ 0, a^s_i ≈0 for alli >0)

Our problem is to identify which satellite is in which scenario. More specifically, the task is to assign a set of metrics {γ^s}, which indicate whether a satellite is likely to be in scenarios (a) or scenarios (b) and (c) (obviously scenario (d) is easily detected by a GNSS receiver as part of its regular operation). Note that we followed [1] in defining multipath and NLOS scenarios separately. However, the treatment applied to both scenarios is the same in this paper.

To tackle the multipath discrimination problem, we propose a setup that is designed specifically for this purpose and that is endowed with the following three features:

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1) A special receiver/antenna configuration with 5 elements (more on this to follow).

2) All the receivers are synchronized (i.e., they use the same clock signal).

3) An attitude estimate with good quality is available from an attitude filter or an IMU.

The proposed receiver antenna array (to be mounted on top of the vehicle) is depicted in Fig.2. We require that the differential of the baselines to satisfy0≤∆_d=|d12−d₂₃| ≤λ/2 and0≤∆_v=|d14−d₄₅| ≤λ/2. For simplicity, and without loss of generality, in this paper, we use d₁₂=d₁₄, d₂₃=d₄₅ (consequently,∆_h= ∆_v= ∆_d). Furthermore, we assume that the baselines in the directions b_x and b_y are perpendicular.

These specific choices are not necessary for the proposed technique to work, but they greatly simplify the explanation of the proposed method. The merits of the configuration in Fig. 2 are discussed in [25], [27] and the references therein.

The major advantages of this configuration are as follows:

1) It allows for GNSS ambiguity resolution and attitude determination to be performed in a simple and computationally efficient manner.

2) It provides attitude estimates with reasonable accuracy even with a small number of visible satellites.

These two features are essential for the proposed multipath detection approach to be computationally feasible as it requires attitude computation to be performed repeatedly using subsets of all the available satellites.

At a certain time epoch, a path of signal arrival from satellite s to antenna i can be described by the signal amplitude a^s_i, the signal phase θ^s_i, and the unit vector of direction of signal arrivalq^s_i. For example, the direct path is described by the triplet (a^s₀, θ₀^s,q^s₀ = −h^s), where h^s is the LOS vector pointing from the receiver to the satellites. Assuming that the combination of signals at each antenna is additive, it can be shown that, for satellite s, the observable phase at antenna 1 is given by [30], [31]:

ϕ^s₁= 1

2πarctan

" PN_{M P}^s

i=0 aisinθ^s_i P^NM P^s

i=0 a_icosθ_i^s

#

, (1)

whereϕ^s₁∈[−0.5,0.5) (cycle).

Further, the phase observations at antennas 2 and 3 are given by:

ϕ^s_r= 1

2πarctan







N_{M P}^s

X

i=0

aisin θ^s_i −2π

λ d1rq^s_i ·bx

N_{M P}^s

X

i=0

aicos θ_i^s−2π

λd1rq^s_i·bx





 , (2)

Wherer∈ {2,3}andλis the wavelength. Note that the phase observations at antennas 4 and 5 have similar formulae, with bx replaced byby.

The observed phase values from Eqs. (1) and (2) are restricted to the range[−0.5,0.5). The whole number of cycles corresponding to each receiver-satellite pair is unknown and is

referred to as the integer ambiguity, which must be estimated before we can make use of the phase measurements. Usually, the phase observations are differenced across receiver pairs to form single-difference observations, and then across satellite pairs to formdouble-differenceobservations. Ambiguities are transformed by these difference operations to form new ambiguities to be estimated. In this work, receiver synchronization is assumed which allows us to take advantage of the single- difference observations where the effect of ionospheric, tropospheric and satellite clock errors is effectively reduced. The receiver synchronization also guarantees that the ambiguities are integers with a between-receiver single difference.

In antenna arrays with an inter-element spacing of less thanλ/2, the single-difference phase ambiguities are all equal to zero. For applications such as attitude determination, the antenna spacing needs to be increased well beyond the λ/2 limit to obtain adequate angular resolution.

To conclude this section, we state the problem of multipath detection using the antenna configuration in Fig.2as follows.

Given the phase observations{ϕ^s₁, ϕ^s₂, ϕ^s₃, ϕ^s₄, ϕ^s₅}^N_s=1^{V S}, where NV S is the number of visible satellites, we want to decide for each satellites∈ {1,· · ·NV S}whether the corresponding phase is contaminated with multipath or not.

III. THEPROPOSEDMULTIPATHDETECTION AND

MITIGATIONMETHOD

In typical applications, the receiver has knowledge (at least approximately) of its location. Given that GNSS satellites are very distant, the satellite LOS vector is known. A signal corrupted with indirect signal arrivals will most likely appear to come from a different (incorrect) direction. By comparing the known satellite LOS and the "apparent" signal arrival direction, we can probably obtain an indication of the presence/absence of the multipath. This intuition, however, fails when practically tested. The nonlinear relation between the direction of arrival and the phase quantities (Eqs. (1) and (2)), followed by the nonlinear computations to disambiguate the phase difference to obtain the apparent direction of arrival, make this method very sensitive to noise.

Another quantity that builds on this intuition is the plat- form’s attitude. Unlike the direction of arrival, which is computed for each satellite independently, attitude is computed from a set of satellite measurements used collectively. If we compute attitude from a subset of satellites that are not affected by multipath, we can likely obtain more accurate attitude estimates compared to when the estimation includes additional multipath-contaminated satellites. This is the main principle upon which our proposed multipath detection approach is built, as will be described in SectionIII-B.

A. Attitude Determination

Our proposed multipath detection and mitigation utilizes the receiver configuration depicted in Fig. 2. To estimate the attitude matrix given phase observations from a certain number of satellites, we follow the same single-difference approach described in [25]. To reduce the complexity of

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the proposed multipath detection and mitigation method we consider only the unconstrained least-squares (LS) attitude estimation approach from [25]. This approach can be summarized as follows. AssumingNksatellite phase observations are available, and collecting all phase measurements for receiver i in a vectorϕi, the single-difference observations model is given by:

ϕij =dijHkb+nij+e, (3) whereHk∈R^N^k^×3is the line of sight matrix forksatellites, b∈ {bx,b_y}is the appropriate baseline vector,n_ij ∈R^N^k^×1 integer ambiguities vector and eare the random errors. After performing a satellite-by satellite ambiguity resolution and refining the results using all the satellite observations collectively adisambiguatedphase vectorϕˆ_ij is obtained. The final estimates of the baseline vectors are computed using LS as:

bˆ_x= (H^TH)⁻¹H^Tϕˆ₁₃,bˆ_y= (H^TH)⁻¹H^Tϕˆ₁₅ (4) Finally, the attitude matrix estimate (from k satellites) can be obtained from the estimated direction vectors (after proper normalization) by:

Rˆk =ˆbx ˆby ˆbx×ˆby

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For completeness, the method in [25] is restated as Algo- rithm 1. Readers interested in the details of the derivations of this method are referred to [25]. Compared to the commonly used attitude determination methods, this method offers a very competitive computational speed. It also performs reasonably well when only very few satellite observations are used.

Notably, it can deliver results using only 3 satellites.

B. Attitude Validation Method

We start with the assumption that the vehicle’s attitude matrixRis known, or at least, a reasonably accurate estimate of it is available. It stands to reason that if a set of phase readings are multipath corrupted, then the attitude computed from these readings will be erroneous by a considerable amount. Consider the following: what if we select from the visible satellites set {1,· · · , NV S}a subset Sk (with at least 3 satellites) and compute the attitude matrix using phase observations from this selected subset Rk? If the computed attitude calculated is very "far" from the known attitude, then we can say that it is more likely that at least one satellite in the set Sk is corrupted with multipath. By examining the attitude matrices estimated using different satellite subsets, we hope that we can identify satellites with significant multipath contributions.

Two questions arise: (1) How do we define the distance between the computed rotation matrix and the known one? (2) Which subsets of satellites do we use for attitude calculations?

To answer the first question, we note that orientation (rotation) matrices form the Special Orthogonal group ofSO(3), which is a Riemannian Manifold [32], with an associated metric between its elements. This metric is known as the geodesic distance. We will use this whenever a distance between two

Algorithm 1 Attitude calculation from phase measurements Function: ATTITUDEFROMPHASE

Inputs: H∈R^N^k^×3,

ϕ1,ϕ2,ϕ3,ϕ4,ϕ5∈R^N^k^×1, d₁₂, d₂₃, d₁₄, d₄₅∈R

1: bˆ_x=ESTIMATEBASELINE(ϕ₁,ϕ₂,ϕ₃, d₁₂, d₂₃,H)

2: bˆ_y=ESTIMATEBASELINE(ϕ₁,ϕ₄,ϕ₅, d₁₄, d₂₅,H)

3: Rˆk =ˆb_x ˆb_y bˆ_x×ˆb_y Output: Rˆk

Function: ESTIMATEBASELINE

Inputs: H∈R^N^k^×3, ϕ₁,ϕ₂,ϕ₃∈R^N^k^×1, d₁₂, d₂₃∈R

1: Kset={−1,0,+1}

2: ∆d← |d23−d12|

3: ϕ12=ϕ1−ϕ2,ϕ23=ϕ2−ϕ3,ϕ13=ϕ1−ϕ3 4: ∆ϕ=ϕ23−ϕ12

5: K= argmin_k∈K_set|(d12/∆d) (∆ϕ+k)|

6: x0= (λ/∆d) (∆ϕ+K) b0= (H^TH)⁻¹H^Tx0 7: N12=Round

(d12/λ)H^Tb0−ϕ12

8: x1= (λ/d12) (ϕ12+N12) b1= (H^TH)⁻¹H^Tx0 9: N₁₃=Round

(d₁₃/λ)H^Tb₁−ϕ₁₃

10: x_F = (λ/d₁₃) (ϕ₁₃+N₁₃) b_F = (H^TH)⁻¹H^Tx₁

Auxiliary Outputs:K,N₁₂,N₁₃ Output: b_F

such matrices is needed. With logm being the matrix loga- rithm, and∥·∥_M being the matrix norm, the distance between two attitude matrices is given by [33]:

d(R₁,R₂) =

logm (R^T₁R₂)

M. (6) To answer the second question, we can first consider all satellite combinations with N_Smin or more satellites (N_Smin≥3). The total number of possible combinations with N_Smin satellites or more and with N_{V S} visible satellites is given by:

Ncomb=

N_{V S}

X

k=N_Smin

NV S

k

(7) which increases combinatorially: with increase in N_{V S} and decrease in NSmin. Hence, using only a select number of combinations is necessary to make the algorithm feasible when the number of satellites is large. We note that this is not a new problem but rather a variation of the maximum coverage problem (a well-known problem in combinatorics) of selecting the best subset from a big set. We shall not dwell on this further.

For simplicity, we propose to select NSmin such that the number of satellite combinations given by Eq. (7) is within computational resources. For example with NV S = 7 using

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Algorithm 2 Satellite Subsets selection algorithm Function: ATTITUDEVALIDATIONDISCRIMINATOR

Inputs: H∈R^N^{V S}^×3,R∈R^N^{V S}^×3, ϕ1,ϕ2,ϕ3,ϕ4,ϕ5∈R^N^{V S}^×1, d₁₂, d₂₃, d₁₄, d₄₅∈R,

N_Smin∈[3, N_{V S}],ϵ_Clus∈R form=N_{V S} downto N_Smin do

fork= 1 to ^N_m^{V S} do

S_k = GETCOMB(N_SV,m,k) Hk=H(Sk,:)

ϕkj=ϕj(Sk,:)for j= 1. . .5

Rmk= ATTITUDEFROMPHASE(Hk,{ϕkj}⁵_j=1,. . .) cmk=d(R,Rk)

end for end for

Clus, cmk = DBSCAN([R,{Rmk}],ϵClus,. . .)

Select cluster with mininal cmk and selectSk∈Cluswith largest|Sk|.

γ_AV^s = 0ifs∈S_k,1 otherwise Output: γ_AV^s ∀ s ∈[1, N_{V S}]

N_Smin= 4will giveN_comb= 64combinations. GivenN_comb satellite combinations, how do we select the best combinations to detect multipath? Consider two ranges of SNR: At higher SNR values (lower phase noise), all combinations not having the multipath corrupted satellites will have good attitude estimates and hence have low error the difference between them is due to noise and geometry-related issues (not related to multipath). On the other hand, at lower SNR values the rotation error due to noise might be approaching the error due to multipath. Thus blindly selecting the minimal error combination is not the best strategy on both sides of the SNR range.

In this work, we cluster the combinations according to their error and then select the "minimum error" cluster. We then select the combination with the largest number of satellites in the "minimum error" cluster. The clustering algorithm we use is DBSCAN [34]. The DBSCAN algorithm requires a cluster distance threshold ϵClus. We note that in typical applications a GNSS receiver can report SNR andϵClus can be based on it. In our simulations, we set this to 10 times the phase noise.

This is summarized in Algorithm 2.

IV. SIMULATIONSETUP& RESULTSDISCUSSION

To test the proposed method, simulations were carried out using the GPS constellation at location 24°28’23.7"N 39°36’38.8"E, in Almadinah AlMunawarah, Saudi Arabia, on Dec 8^th, 2022 at 16:05:00 UTC time. A cutoff elevation angle of 30^◦ is applied to simulate the effect of building in urban areas. This mimics the effect of being enclosed in a circular area surrounded by buildings or walls of the same height. The satellite sky-plot using a cutoff elevation angle of 30^◦ is shown in Fig. 3, with a number of visible satellites N_{V S} = 7. In our simulations, we consider only the L1 frequency (λ₁ ≈ 19.05cm). Baseline lengths are chosen

N

30°

60°

E

120°

150°

S 210°

240°

W 300°

330°

0° 20° 40° 60° 80°

PRN-1

PRN-2 PRN-3

PRN-10

PRN-19

PRN-27 PRN-31

Fig. 3. Simulated Sky plot at 24°28’23.7"N 39°36’38.8"E on Dec 8^th, 2022 at 16:05:00 UTC

as d12 = 5λ1/2, d23 = 5.9λ1/2d13 = 10.9λ1/2, giving a differential baseline length of ∆d= 0.45λ1.

The noise-free phase observations are generated using Eqs. (1) and (2). White Gaussian noise with a standard deviationσϕis added to the phase and the results are wrapped to the interval[−0.5,0.5). The number of multipath-affected satellites is set asN_{M P} = 1,2 or3, with the affected satellites chosen randomly from all visible satellites. The multipath- affected signal is given a random amplitude between 0.5 and 0.8. This is based on the assumption that much weaker multipath signals can be adequately dealt with by the previous receiver processing stages. The multipath reflection points are selected randomly in the vicinity of the receiver.

Using the described setup parameters, we perform 5000 simulation trials to compute 4 performance metrics:

1) The (overall) miss-classification (MC) rate, which is the total number of miss-classifications divided by the total number of entries (5000×NV S). A miss-classification is defined as a multipath-affected satellite being incorrectly detected as multipath-free or vice versa. See Fig.4(a).

2) The false-negative rate: the number of incorrect classifications of a multipath-affected satellite as a multipath- free satellite divided by the number of multipath affected satellites (5000×N_{M P}). See Fig. 4(b).

3) The false-positive rate: the number of multipath-free satellites incorrectly classified as multipath-affected divided by the number of clean satellites (5000×(NV S− NN M P). See Fig.4(c).

4) Success Rate: the number of times the algorithm was able to correctly identify all the multipath-affected and all the multipath-free satellites correctly divided by the total number of trials (5000). See Fig. 4(d).

As can be expected, we can see from Fig. 4 that miss- classification rates and success rates clearly depend on the phase noise level. For example, the false-negative rate starts at close to 10% when 1 or 2 satellites are affected by multipath,

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1 1.5 2 2.5 3 3.5 4 4.5 5 Phase Noise

0 5 10 15 20 25

30 Misclassification Rate (FP and FN) (%) NMP=1

NMP=2 NMP=3

1 1.5 2 2.5 3 3.5 4 4.5 5

Phase Noise 0

10 20 30 40 50

60 False Negatives (%)

NMP=1 NMP=2 NMP=3

1 1.5 2 2.5 3 3.5 4 4.5 5

Phase Noise 0

2 4 6

8 False Positives (%)

NMP=1 NMP=2 NMP=3

1 1.5 2 2.5 3 3.5 4 4.5 5

Phase Noise 0

20 40 60 80

100 Success Rate (%)

NMP=1 NMP=2 NMP=3

Fig. 4. Miss-classification Rates as a function of phase noise standard deviation: (a) Overall Miss-classification Rate (b) false-negative Rate (c) Clean Satellite Miss-classification Rate (d) Success Rate. Note that Black, Red and Blue are 1, 2 and 3 multipath affected satellites respectively.

1 1.5 2 2.5 3 3.5 4 4.5 5

Phase Noise 10^-1

10⁰ 10¹ 10²

Mean Absolute Error (°)

NMP=1 All. Sats NMP=2 All. Sats N_MP=3 All. Sats

NMP=1 Sel. Sats NMP=2 Sel. Sats N_MP=3 Sel. Sats

Fig. 5. Mean Absolute Error in X and Y baseline as compared to the ground truth for 1, 2 and 3 multipath errors as a function of phase noise

and increases as the phase noise level increases. We can also observe that the accuracy of the proposed algorithm in identifying multipath/non-multipath scenarios decreases as the number of multipath-affected satellites increases. We note that the proposed method can only deal with scenarios with a number of multipath-affected satellites N_{M P} ≤N_SV −3, as we cannot compute the attitude matrix with less than three multipath-free satellites.

Fig.5shows the errors in degrees of the estimated x and y axes direction vectors when using only the satellites detected by the algorithm as multipath-free (red) and when using all visible satellites (black). The error is sufficiently small at 1- 3mm phase noise (≈ 1^◦) and reaches higher levels close to

10^◦ when the phase noise reaches 5mm. Compared to using all the available satellites, using the reduced satellite set offers a significant advantage.

So far our performance evaluation considered that a per- fectly known attitude matrix is known at the start. In practice, the attitude matrix is not known with perfect accuracy. Instead, it is contaminated with error contributions from attitude determination using GNSS-based methods [35]–[37] and/or IMU [38]. We analyze this by simulating the case where the algorithm is fed a perturbed orientation matrix. The attitude matrix of the receiver is converted to Euler angles, and a random perturbation uniformly distributed between [−U/2, U/2]^◦ is added to all angles. We fix a phase error of 2mm and re- evaluate various performance metrics. The results are shown in Fig. 6(a)-(d). As expected, the ability to detect multipath correctly decreases as the rotation matrix error gets worse. Per- formance also deteriorates as the number of multipath-affected satellites increases. The only result that seems slightly surpris- ing is in Fig.7(b), where the false-negative rates appear to be lower for larger numbers of multipath-affected satellites. We note that this happens at relatively higher miss-classification rates and hence might involve greater dependence on chance and randomness.

Finally, Fig. 7 shows the error in the baselines direction vector estimation as the rotation matrix noise increases. We can see that the proposed algorithm still offers a significant error reduction compared to the naive algorithm that uses all the satellite observations.

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0 5 10 15 20 25 30 35 40 Rotation Noise ( °)

5 10 15 20 25 30

35 Misclassification Rate (FP and FN) (%) NMP=1

NMP=2 NMP=3

0 5 10 15 20 25 30 35 40

Rotation Noise ( °) 30

40 50 60 70

80 False Negatives(%)

NMP=1 NMP=2 NMP=3

0 5 10 15 20 25 30 35 40

2 4 6 8

10 False Positives(%)

NMP=1 NMP=2 NMP=3

0 5 10 15 20 25 30 35 40

20 30 40 50 60

70 Success Rate (%)

NMP=1 NMP=2 NMP=3

Fig. 6. Miss-classification Rates as a function of rotation noise withσϕ= 2mm (a) Overall Miss-classification Rate (b) false-negative Rate (c) Clean Satellite Miss-classification Rate (d) Success Rate. Note that Black, Red, and Blue are 1, 2, and 3 multipath-affected satellites respectively.

0 5 10 15 20 25 30 35 40

Rotation Noise ( °) 10⁰

10¹ 10²

Mean Absolute Error (°)

NMP=1 All. Sats NMP=2 All. Sats N_MP=3 All. Sats

NMP=1 Sel. Sats NMP=2 Sel. Sats N_MP=3 Sel. Sats

Fig. 7. Mean Absolute Error in X and Y baseline as compared to the ground truth for 1, 2, and 3 multipath errors as a function of rotation noise

V. CONCLUSIONS& FUTUREWORK

We have presented a method to determine whether GNSS satellite phase observations are affected by multipath. The proposed method takes advantage of a special receiver configuration to detect multipath using an attitude validation approach.

Through simulations, we have demonstrated the proposed method’s performance in detecting multipath-affected satellite observations.

Further work is needed to evaluate the suitability of the algorithm in various practical scenarios. That includes:

1) Experimental validation of the algorithm.

2) Applying the algorithm to GNSS positioning using phase observations with possible inclusion of pseudorange ob-

servations.

3) Designing an attitude tracking filter that consolidates both GNSS and IMU observations to provide a good initial attitude estimate for the proposed algorithm to start from.

4) Modify the algorithm to eliminate the need for an external attitude estimate.

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