THE key advantages of the Global Positioning System. Robust GNSS Multi-receiver Direct Position Estimation for High-dynamic Aerial Applications

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1 1 Robust GNSS Multi-receiver Direct Position Estiation for High-dynaic Aerial Applications Arthur Hsi-Ping Chu, Student Meber, IEEE, and Grace Xingxin Gao, Senior Meber, IEEE Abstract To itigate the robustness challenges (e.g. ultipath, asking) posed to legacy GPS receiver architectures, prior work has proposed Direct Position Estiation (DPE), a new, robust GPS receiver architecture that preserves inforation redundancy by directly estiating the navigation solution in the position doain, skipping interediate ranging easureents altogether. This paper proposes ulti-receiver Direct Position Estiation (MR-DPE), a novel GPS receiver architecture that operates ultiple DPE receivers over fixed antenna baselines as a receiver network. This network aggregates the easureents taken at individual receivers to for a unified position-velocity-tie (PVT) solution. MR-DPE further expands DPE s robustness through the easureent and geoetric redundancies resulting fro the ulti-receiver architecture. We ipleent MR-DPE on a software-defined radio and conducted full-scale flight experients. We installed four GPS L1 active antennas on a high-dynaic fixed-wing aircraft and explored a wide range of flight profiles, especially those under significant signal ultipath and asking effects. The flight test has validated the superior robustness perforance of MR-DPE. Index Ters Global Navigation Satellite Syste (GNSS), Global Positioning Syste (GPS), Direct Positioning, Maxiu Likelihood Estiation, GNSS receiver I. INTRODUCTION THE key advantages of the Global Positioning Syste (GPS) over other navigation technologies include its worldwide coverage [1] and its infrequent need for receiver calibration, if ever. These unique characteristics ake GPS highly suitable for airborne platfors [2], whose operation is traditionally constrained by land-based radio beacons and the drifting accuracy of inertial navigation equipent [3], [4]. Under an ideal signal environent, a legacy GPS receiver estiates the ranges between its antenna and each visible GPS satellites. Based on these ranging easureents, the receiver uses triangulation to for a position-velocity-tie (PVT) solution [1], [5]. However, such receiver architecture bears weaknesses when the signal environent is not ideal [6] [8]. The robustness of a receiver is the receiver s ability to aintain its accuracy under non-ideal signal environents. For instance, when a satellite s signal experiences ultipath, a non-line-of-sight (NLOS) coponent will join the original line-of-sight (LOS) signal. This NLOS coponent is a delayed and attenuated shadow of the LOS one that blurs the ranging easureent and propts larger errors [6], [7], [9]. Siilarly, Arthur H Chu and Grace Xingxin Gao are with the Departent of Electrical and Coputer Engineering, University of Illinois at Urbana-Chapaign, Urbana, IL 61801, USA. E-ail: achu11@illinois.edu, gracegao@illinois.edu. when a satellite s signal is weakened by asking effects, the noise level in the easureents rises. This rising noise level renders the easureents ore error-prone, if not copletely undetectable. The call for robustness is particularly iperative for receivers installed on airborne platfors, which tend to travel at higher speeds and engage in ore dynaic otion profiles. A faulty navigation solution hence has grave iplication, as it ay not only ipact noral operation but create iinent safety hazard [10]. We choose to focus on ultipath and asking effects in this paper as they occur ore frequently than other fors of robustness challenges, such as jaing and eaconing. Existing works that focused on GPS receiver robustness had explored a yriad of approaches [8], [11]. [12] [16] proposed Kalan-Filter-based vector tracking loops that jointly track ranging easureents across different satellites (i.e. channels ). [17] [21] fused GPS receivers with inertial easureent units (IMUs) to provide uninterrupted service even when the receivers underwent robustness challenges. [22] [26] deployed various carrier-level techniques to enhance receiver robustness. [16], [27] installed ore than one GPS receivers onto the sae platfor to explore the integration of easureents for enhanced easureent and geoetry redundancy. [28] proposed new decoding schee that iproved signal-tonoise ratio (SNR). Last but not least, [29] [40] discussed Direct Position Estiation (DPE), which directly estiates navigation solutions in the position doain, eliinating the need for interediate ranging easureents. This approach creates a deeper coupling across different channels and utilizes weak signal that would have otherwise been discarded. In this paper, we propose ulti-receiver Direct Position Estiation (MR-DPE), which is a significant expansion fro our preliinary investigation into MR-DPE [41]. MR-DPE is a novel GPS receiver architecture that develops fro singlereceiver DPE (SR-DPE), [30], [40]. In [41], the backbone architecture of MR-DPE was briefly discussed, along with soe early results fro the flight test. The novelties that set this paper apart fro [41] include a new, DPE-inspired attitude estiation algorith, a ore thorough atheatical/algorithic analysis, and ore experiental results. In addition, we copare the outputs of MR-DPE against those of a reference syste ounted on the aircraft. The syste provides highly accurate PVT inforation using IMU and DGPS with a ground reference station. MR-DPE extends SR-DPE with a ulti-receiver architecture to gain additional inforation redundancy. We organize

2 2 Fig. 1. A siplified exaple for geoetric redundancy. Two GPS antennas (arked by the solid ovals) are ounted on both wingtips of a fixed-wing aircraft. Both of the have respective blind spots caused by the fuselage, in which sky view is being blocked. Aggregating the easureents of the two antennas will help to create a coplete view of the sky. these receivers into a receiver network that generates a unified, network-wise PVT solution. MR-DPE ais at iproving receiver robustness in 1) signal asking and 2) ultipath environents, which is achieved via three ajor aspects: 1) On the receiver level, MR-DPE inherits robustness fro individual DPE receivers. [30], [35], [36], [40] illustrated how DPE considerably enhanced receiver robustness when the noise level was high or when ultipath effect was pronounced. 2) On the network level, since the baselines aong the antennas are fixed, we associate one receiver s easureents with others through linear transforations. This creates a easureent redundancy. 3) One of the ost proinent application for MR-DPE, as priarily investigated in this paper, is when the antennas are installed onto different parts of a single, rigid platfor. While each of these antennas ay have a different, incoplete sky-view, we create a ore coplete observation by aggregating the inforation these antennas respectively capture. This geoetric redundancy facilitates a yet higher level of robustness. Fig. 1 uses a siple exaple to illustrate this benefit. In order to evaluate the robustness perforance of MR- DPE, we conducted full-scale flight tests on a high-dynaic, fixed-wing aircraft. The flight tests distinguish this paper fro all of the prior DPE works, whose experients have been on static settings [34], on land-based platfors [37], [40], or on siulations [31], [34] [36], [38]. We recorded in-flight raw GPS data for analysis, with focus on scenarios where ultipath and signal asking are likely to occur. For instance, in one of the test points, the aircraft flew in a river valley, below the ridgelines, with the heights of the surrounding terrain exceeding the aircraft altitude by ore than 500 eters. We process the data using our MR-DPE ipleentation on a softwaredefined radio, and copare its robustness perforance against existing architectures, including scalar tracking and singlereceiver DPE. The reainder of the paper is organized as follows. Section II recaps the algorith used in the individual DPE receivers, and we then ove on to provide technical details for MR- DPE in Section III. Section IV describes the ipleentation we choose for MR-DPE, as well as the experient setting we used to validate MR-DPE. The results of the experients are presented in Section V. Finally, Section VI suarizes the paper. II. DIRECT POSITION ESTIMATION In this section we briefly discuss the operation of a single, standalone DPE receiver, along with soe GPS positioning fundaentals. In a high-level sense, DPE holds the sae operation principle as any other GPS receiver; that is, it akes an observation to the raw GPS signal Y, and uses that observation to estiate a PVT coordinate, X, for the receiver s antenna. X x ẋ c X (i) ρ (i) ρ (i) Y (i) Y D (i) : Receiver PVT coordinate = [x, y, z, cδt, ẋ, ẏ, ż, c δt] = [x, ẋ] (1) : Receiver position and clock bias () = [x, y, z, cδt] (2) : Receiver velocity and clock drift (/s) [ = ẋ, ẏ, ż, cδt ] : Speed of light, in vacuu (/s) = (3) : i-th [ visible satellite PVT coordinate = x (i), y (i), z (i), cδt (i), ẋ (i), ẏ (i), ż (i), cδt (i)] (4) : Pseudo-range for i-th visible satellite () = x x (i) (5) : Pseudo-rate for i-th visible satellite (/s) ( ) x x (i) = (ẋ ẋ (i) ) (6) ρ (i) : GPS signal ( odel for i-th visible satellite ) = D (i) G (i) 2π(f (i) code t + φ(i) code ) (i) j2π(f e carr t+φ(i) carr ) (7) : GPS signal odel for all visible satellite L = Y (i) (8) i=1 : Data bit for i-th visible satellite G (i) ( ) : L1 C/A code for i-th visible satellite f (i) code φ (i) code f (i) carr φ carr (i) f C/A f L1 : Code frequency for i-th visible satellite (Hz) = f C/A f C/A ρ (i) c : Code phase for i-th visible satellite (rad) = f C/A ρ (i) c : Carrier frequency for i-th visible satellite (Hz) = f L1 f L1 c ρ(i) : Carrier phase for i-th visible satellite (rad) : GPS coarse acquisition code chipping rate (Hz) = : L1 carrier frequency (Hz) = L : Nuber of visible satellites (i = 1,..., L) Equations (1) to (8) indicate that the observed GPS raw signal Y is a function of the receiver state, X, and the states

3 3 Fig. 2. 2D exaple for DPE candidate grid, ˆX, and its likelihood distribution, P( )[42] of the visible satellites, X (i). Furtherore, as the satellite coordinates X (i) is retrievable fro the epheerides, the signal Y can be expressed as: Y = Y (X) (9) That is, the raw signal is a of the receiver s PVT coordinate. This stateent is the very foundation of GPS positioning. Now, instead of deriving X fro the paraeters of Y (e.g. φ code, f carr ), a DPE receiver sets up a grid of candidates, ˆX. Each of the DPE candidates 1) is a unique cobination of [x, y, z, cδt, ẋ, ẏ, ż, cδt] coordinates and 2) represents a potential solution for X. The goal of the DPE receiver is to evaluate the likelihood of each candidate being the navigation solution. Fig. 2 depicts a siplified, 2D candidate grid, coplete with a likelihood anifold that illustrates the distribution of the aforesaid likelihood. Note that, since the solution is to be generated fro this grid, it is iportant for the grid to actually cover the solution. For instance, on x-axis: in ˆx x ax ˆx (10) The sae should hold true for all eight diensions. An initialization is therefore necessary. In DPE, we can obtain the initializing coordinates either by a scalar tracking receiver, a heterogeneous navigation syste (e.g. IMU), or the user s anual input. These coordinates do not have to be perfect or very accurate, as long as the grid will satisfy the condition of equation (10). Once the candidate grid ˆX is in place, the DPE receiver generates a coposite signal replica, Ŷ, for each of these candidates, following the recipe prescribed by equations (1), (5), (7) and (8). The replica is called coposite because it is the superposition of the signal replicas of all L visible satellites. That is: ˆX : -th PVT coordinate candidate [ = ˆx, ŷ, ẑ, c ˆδt, ˆẋ, ˆẏ, ˆż, cδt ˆ ] (11) Ŷ (i) Ŷ : Signal replica for i-th visible satellite, w.r.t. -th candidate : Coposite replica for -th candidate L = Ŷ (i) (12) i=1 This replica essentially eulates the signal reception fro the entire GPS constellation at the candidates coordinate. The DPE receiver uses the correlation between each Ŷ and the raw GPS signal Y to deterine the likelihood for the -th candidate being the navigation solution. We define the likelihood function, P, with respect to the candidate coordinates, as: P( ˆX ) = corr(ŷ, Y ) (13) Depending on ipleentation, the DPE receiver can use different schees to deduce a navigation solution X fro the likelihood anifold P( ). For instance, it ay select the candidate with the axiu likelihood as the navigation solution; that is: X = arg ax P( ˆX ) (14) ˆX The receiver ay also ipleent a weighted-average approach: M =1 X = P( ˆX ) ˆX M =1 P( ˆX (15) ) In this paper, we follow our prior work on the efficient DPE receiver ipleentation in [40]. The ost significant distinction between [40] and the generic DPE receiver defined above is that it decouples the search space for X, into two separate search spaces for x and ẋ. This approach has been proven to provide considerable iproveent over tie coplexity for the otherwise coputationally intensive DPE algorith. [40] delivered significant iproveent in robustness over legacy architectures, while the decoupling process posed no discernible ipact on the accuracy. The detail of this decoupling process is beyond the scope of this paper. It suffices to know that, instead of one likelihood anifold P( ˆX ) for the full navigation solution, a DPE receiver under this ipleentation would generate two anifolds: one for position/clock bias x and the other for velocity/clock drift ẋ. That is: ˆx ˆẋ n : -th position/clock bias candidate = [ˆx, ŷ, ẑ, c ˆδt ] : n-th velocity/clock drift candidate = [ˆẋ n, ˆẏ n, ˆż n, cδt ˆ n ] R(ˆx ) : Likelihood function for -th position and (16) clock bias candidate (code correlogra) = corr(y, Ŷ,code) F(ˆẋ n ) : Likelihood function for n-th velocity and (17) Ŷ,code clock drift candidate (carrier spectru) = F(Y Ŷ n,carr) : Code-only coposite replica for -th candidate

4 4 Ŷ n,carr = L i=1 G (i) (i) (i) ( ˆf,codet + ˆφ,code ) (18) : Carrier-only coposite replica for n-th candidate L ( ) (i) (i) = exp j2π( ˆf n,carrt + ˆφ n,carr) (19) i=1 The receiver state is then estiated: x = R(ˆx ) ˆx R(ˆx (20) ) n ẋ = F(ˆẋ n ) ˆẋ n n F(ˆẋ (21) n ) X = [x, ẋ] (22) III. MULTI-RECEIVER DIRECT POSITION ESTIMATION In MR-DPE, we organize K DPE receivers into a receiver network. This receiver network is designed to operate with stationary GPS antenna geoetry; that is, the antenna baselines, bk, are fixed with respect to the local coordinate frae (e.g. East-North-Up frae, ENU) of the network. Hence, while each of the receivers akes a different GPS signal observation, Y k, the GPS easureents they ake (i.e. likelihood anifolds R k and F k ) can be projected through the local frae with linear transforations, such as rotations and translations. In MR-DPE, we aggregate the easureents at the centroid, O, of the network, which is also defined as the origin of the local coordinate frae. We use these aggregated easureents to generate a unified navigation solution, X o. Fig. 3 depicts such an exaple for the easureent aggregation process, where the position anifolds R k of the individual receivers are projected toward the centroid. A rigid platfor, such as the fixed wing aircraft depicted in the exaple, provides the fixed baselines essential for our MR- DPE algorith. We observe that the anifolds will overlap each other and lead to a unified easureent R o. Fig. 3 also highlights the need for MR-DPE to obtain the attitude of the network (platfor), which is the tether between the global coordinate frae (in which easureents are ade at individual DPE receivers) and the local frae (in which the baselines b k are defined and projections are ade). A. Network PVT Estiation Fig. 4 shows the architecture of our MR-DPE algorith. For the purpose of clarity, the step-wise insets (i)-(iv) reuse the exaple introduced in Fig. 3, where four color-coded antennas are installed on the wing tips (left-red, right-green), nose (orange) and tail (blue). The MR-DPE algorith operates iteratively to estiate the network s PVT coordinate. Each iteration is broken down as follows: 1) Initialize network PVT coordinate, X o, and attitude, α, where X o : Network PVT state = [x o, y o, z o, cδt o, ẋ o, ẏ o, ż o, c δt o ] Fig. 3. An exaple for aggregation of position likelihood anifolds, R k. They are projected to the centroid of the network, O, with the knowledge of antenna baselines b k and aircraft yaw, pitch and roll (the latter two not shown). x o ẋ o α = [x o, ẋ o ] : Network position/clock bias = [x o, y o, z o, cδt o ] : Network velocity/clock drift = [ẋ o, ẏ o, ż o, cδt o ] : Network attitude: yaw, pitch and roll = [ϕ, θ, φ] They can be initialized in a variety of fashions, including a) values fro previous iterations, b) anual input, c) scalar tracking, etc. 2) Generate the position and velocity candidate grids, ˆx o, and ˆẋ o,n, where ˆx o, ˆẋ o,n : Network position/clock bias candidate, -th = [ˆx o,, ŷ o,, ẑ o,, c ˆδt o, ] : Network velocity/clock drift candidate, n-th = [ˆẋ o,n, ˆẏ o,n, ˆż o,n, c ˆ δt o,n ] This is depicted in Fig. 4 as step (i), where a fraction of ˆx o, candidates are shown as black spheres around the center of the fuselage. Note that, the range of the candidates should reflect the dynaics of the network. For instance, the candidate grids for networks traveling at high velocity should cover wider ranges, prescribed by equation (10). In addition, the density of the grid should be such adjusted that it properly reflects the trade-off between the coputation tie and the output resolution. 3) Project the network s position/clock bias candidate grid, ˆx o, to the individual receivers antennas. This would create K candidate grids, denoted as ˆx k,, each surrounding one of the receivers. We define a transforation function T k (x, α) for this operation, where ˆx k, : k-th receiver position/clock bias candidate, -th

5 5 Fig. 4. MR-DPE block diagra = [T k (ˆx o,, α)] ECEF T k (x, α) = R x (φ) R y (θ) R z (ϕ) ((x) ENU + ) b k bk R O (p) ECEF (x) ENU : k-th antenna baseline, in ENU w.r.t. O (23) : 3 3 rotation atrix around the given axis : Network centroid = [0, 0, 0] ENU : Converting a given local ENU coordinate p = x w.r.t. O to global ECEF coordinate frae : Converting a given global coordinate x = p to local ENU w.r.t. O This step is depicted in Fig. 4 as step (ii). The corresponding inset shows that the black candidate grid previously surrounding the center of the fuselage is now replaced by four colored grids, each surrounding their respective antennas. The network attitude, α, can be derived fro velocity vector [16]. Under this schee, it is assued that the network s travel direction is aligned with its attitude (i.e. it points where it goes ). Another approach, which we propose in Section III-B, uses ultiple attitude candidates and a DPE-inspired approach to find the attitude with the highest likelihood. Last but not least, this inforation ay be provided by an external aiding source, e.g. an inertial easureent unit (IMU). For the velocity grid ˆẋ o,n, we observe that the velocity over different parts of a rigid network is relatively unifor (as it is rigid). We therefore assign ˆẋ o,n directly to each receivers. That is, ˆẋ k,n : k-th receiver velocity/clock drift candidate, n-th = ˆẋ o,n (24) 4) We then evaluate the probability distribution of candidate grids, on a receiver-by-receiver basis. This is alost identical to the single-receiver DPE operation discussed in Section II. The ajor distinction is that each receiver now uses propagated candidates ˆx k, and ˆẋ k,n, instead of the ones generated by the individual receivers. Following the evaluation, we obtain two likelihood anifolds per receiver. They are expressed as: R k (ˆx k, ) : k-th receiver position/clock bias likelihood = corr(y k, Ŷk,,code) (25) F k (ˆẋ k,n ) : k-th receiver velocity/clock drift likelihood Y k = F(Y k Ŷ k,n,carr) : GPS raw signal observed at k-th antenna This is shown as step (iii) in Fig. 4.

6 6 5) We aggregate the anifolds of the individual receivers and create two network-wise anifolds for the position/clock bias and the velocity/clock drift. That is: R o (ˆx o, ) : Network position/clock bias likelihood anifold = R k (ˆx k, ) (26) k F o (ˆẋ o,n ) : Network velocity/clock drift = k likelihood anifold F k (ˆẋ k,n ) (27) This step is shown as step (iv) in Fig. 4. The significance of equations (26) and (27), together with (23) and (24), is that we are fully leveraging the fixed antenna baselines, b k, to associate ultiple easureents through spatial transforation. That is, we obtain ore than one likelihood value (i.e. easureent) for the sae candidate. This creates an inforation redundancy that will contribute to the enhanced receiver robustness. 6) With R o and F o in place, we derive the navigation solution, X o, with the weight-average ethod introduced in equations (20), (21) and (22); that is: X o = [ R o(ˆx o, ) ˆx o, R, o(ˆx o, ) n F o(ˆẋ o,n ) ˆẋ o,n n F o(ˆẋ o,n ) (28) 7) The last step is to tie-propagate the navigation solution, in order to provide initialization for the next iteration. Kalan Filter or other filtering techniques ay be deployed to reject outliers aong inputs and reduce noises in the output. However, with the focus on robustness analysis, a siple, one-step propagation schee will suffice for the scope of this paper. That is: where X(t + T ) = FX(t) (29) T : Propagation tie step F : Propagation atrix, 8 8 = The iteration concludes. B. Attitude Deterination I 4 T I 4 O 4 4 I 4 In this paper we investigate two schees to deterine the network attitude using generic GPS signals, without utilizing external aiding source such as an inertial easureent unit (IMU). 1) Velocity Vector: The first schee follows the assuption that the network tracks the direction it is pointing at, and uses the velocity vector ẋ to derive the attitude of the network. ϕ : Yaw ] θ ẋ ENU = arctan2(ẋ e, ẋ n ) : Pitch ( ) ẋ u = arctan ẋ2 e + ẋ 2 n : Velocity in East-North-Up coordinate frae = [ẋ e, ẋ n, ẋ u ] This schee is applicable for any GPS receiver regardless of architecture, as long as the receiver provides 3D velocity vector as a part of its navigation solution. Previous work has deployed this schee in deterining the heading of a land vehicle [16]. The advantage of this schee lies in its siplicity, which brings coputational efficiency and hence real-tie perforance. The ajor disadvantages include 1) it is ipossible to deterine the roll φ of the network fro ẋ and 2) the assued identity between the heading and the network s pointing direction ore than often does not hold true, especially for aerial platfor. For instance, a fixed-wing aircraft usually exhibits an angle of attack and ay experience side-slips under crosswind conditions. We therefore propose a new attitude deterination schee, as described below. 2) Maxiu-likelihood Attitude Estiation: We propose in this paper a new attitude deterination schee that is custoized for our MR-DPE algorith. This schee starts with odifying step 3 in Section III-A. Instead of using a known α, we create a grid of attitude candidates, ˆα l, where ˆα l : l-th attitude candidate; yaw, pitch and roll = [ ˆϕ l, ˆθ l, ˆφ l ] Siilar to PVT candidates introduced in equation (11), each of these attitude candidate represents a unique cobination of yaw, pitch, and roll values. The attitude deterination algorith proceeds as follows: 1) We generate the attitude candidates to surround an initialization attitude. This initialization ay be obtained fro the previous iteration, an external aiding source, or user s anual input. Note that the condition fro equation (10) reains binding for the attitude candidates. That is, in l in l in l ˆϕ l ϕ ax l ˆθl θ ax l ˆφl φ ax l Therefore, the ranges of these candidates should as well reflect the dynaics of the network. 2) For each of the attitude candidates ˆα l, we repeat step III-A-3 to -5 by replacing equation (23) with: ˆx k,,l ˆϕ l ˆθ l ˆφ l : -th position/clock bias candidate, of k-th receiver, under l-th attitude candidate = T k (ˆx o,, ˆα l ) (30) Fig. 5 illustrates this operation with three attitude candidates of different yaw values (pitch and roll are not shown). As these candidates are applied to the transforation T k, the network candidate grid is projected

7 7 Fig. 5. Exaple for attitude candidates, with ϕ ` = 15, 0, and 15. (a) Matching yaw candidate. Peaks locate consistently across individual DPE receivers. Fig. 6. The effect on Ro,` with real word data, by (is-)atching attitude candidates α `. differently in the local coordinate frae. Note that, only the correct attitude candidate (ϕ = 0) will create receiver PVT candidate grids x k,,` that correctly overlay the ounting positions of the antenna. The other two attitude candidates create receiver PVT candidates that are offset fro the antennas, with each antenna having a different offset values (e.g. for ϕ = 15, the nose and the tail grids offset to the right and the left of the antennas, respectively). 3) Siilar to step III-A-4, we evaluate the likelihood anifolds, Rk,` of individual receivers under a given attitude candidate. It follows equation (25) by replacing x k, with x k,,`. 4) We now generate the network-wise likelihood anifolds, one for each attitude candidate. This follows the forula described in equation (26). That is: X Ro,` (x o,,` ) = Rk,` (x k,,` ) (31) (b) Misatching yaw candidate. Peaks are seen to have different offsets in different individual DPE receivers. Fig. 7. Receiver position/clock bias likelihood anifolds Rk,`. Likelihood peaks are highlighted in red. k Note that the offset of the candidate grids, described above and illustrated in Fig. 5, will lead to the correlation peaks in Rk,` being offset fro one another. This is illustrated in Fig. 6. Fig. 7(a) and Fig. 7(b) use the Rk,` of a real-world data set to highlight the effects of atching and isatching attitude candidates. While the aligned attitude candidate gives out aligned peak positions in all four receivers, the isatching attitude candidate does uch less so. This outcoe is consistent with our theory for this algorith. Fig. 8(a) and Fig. 8(b) depict the aggregated likelihood anifolds Ro,`. It is observed that the offset in Rk,` correlation peaks, caused by the isatching attitude candidate, leads to a uch flatter correlation peak in the network likelihood anifold, with overall lower correlation values. In contrast, the atching attitude candidate gives out a considerably sharper peak. 5) Finally, we deterine the attitude by selecting the attitude candidate that provides the sharpest peak. We develop the following etric to evaluate the sharpness of the peak: S(α ` ) = s X (M Ro,` (x )) 2 (32)

8 8 (a) Matching yaw candidate (b) Misatching yaw candidate Fig. 8. Network position/clock bias likelihood anifold R o,l, for different yaw candidates Architecture Scalar Trk. Vector Trk. DPE Measureent [ρ (i), ρ (i) ] [ρ (i), ρ (i) ] [x, ẋ] Receiver state [ρ (i), ρ (i) ] [x, ẋ] [x, ẋ] TABLE I COMPARISON OF DIFFERENT SINGLE-RECEIVER ARCHITECTURES, ASSUMING L VISIBLE SATELLITES (i.e. i = 1,..., L) where M = ax R o,l(ˆx ) We use this sharpness etric to estiate the network attitude. We ay either directly select the candidate with the highest sharpness value, or use S to weight average the candidates, as previously described for PVT candidates in equation (28). This concludes our attitude estiation algorith. C. Robustness Benefits The robustness benefits of MR-DPE coe in three-fold. First, it inherits robustness fro its base receiver architecture, DPE. The key distinction between a DPE receiver and legacy GPS receiver architectures is that DPE creates estiates directly in the navigation (i.e. PVT) doain, skipping the interediate pseudo-range/-rate easureents. This creates a deep coupling across different channels. In contrast, both scalar vector tracking loops [12], [16] easure the pseudo-ranges/- rates of each satellite, though a certain level of coupling is provided in vector tracking by aintaining the receiver state in the navigation doain. This distinction is highlighted in Table I. Second, under MR-DPE, we aggregate the easureents (i.e. likelihood anifolds) taken at individual receivers and fors a unified, network-wise PVT solution. This creates an over-deterined estiation proble, increasingly ore so as ore DPE receivers are added to the architecture. Table II highlights this reduction in the nuber of unknowns. Last but not least, we have discussed the geoetric redundancy in Section I and Fig. 1. This geoetric redundancy, introduced by the ulti-receiver architecture, will bring an additional layer of robustness to our syste design. IV. IMPLEMENTATION AND EXPERIMENTS We ipleent the MR-DPE algorith on our Pythonbased software-defined radio (SDR) research suite, PyGNSS. Fig. 9. Antenna ounting points on flight-test aircraft PyGNSS facilitates flexible, object-oriented ipleentation of new GPS receiver algoriths, and the verification thereof [16], [43]. PyGNSS takes interediate-frequency raw GPS data as its input, which is collected with coercial offthe-shelf radio front-ends. Each of the front-ends, an Ettus Research TM Universal Software-radio Peripheral (USRP), records data at L1-frequency fro one active antenna installed on the platfor. These USRPs are collectively clocked by a Syetrico R SA.45s chip-scale atoic clock (CSAC). The CSAC provides all front ends with a 10-MHz clocking signal at a very low drift rate (cδt), thus considerably reducing clockoriginated errors. Our experient platfor is a high-dynaic, fixed-wing aircraft, weighing approxiately 12,000 lbs upon take-off. We installed four GPS L1 active antennas onto the aircraft, with one at each wingtip, one in front of the cockpit canopy, and one on top of the epennage. The antenna baselines b k are accurately easured using FARO R FaroAr. Each of these antennas was connected to a CSAC-clocked USRP. Raw data was logged to on-board laptops and later exained with the PyGNSS-ipleented MR-DPE receiver. Fig. 11 shows the 4 USRPs being ounted on an air-worthy equipent rack and installed in the flight test aircraft. Fig. 12 presents the block diagra of our hardware setup. The results are copared against those delivered by legacy scalar tracking receiver as well as single-receiver DPE, which are both ipleented with PyGNSS. In addition, the aircraft was equipped with a Tie-Space-Position-Inforation (TSPI) syste, which generated highly accurate PVT solutions through an IMU and a differential GPS receiver. The TSPI data therefore acts as a truth source against which we can copare the outputs of our MR-DPE ipleentation. The hardware installation of TSPI can also be seen in Fig. 11. V. RESULTS AND ANALYSIS Our test points were selected to exaine MR-DPE s perforance under dense signal asking and ultipath. These test points include operating the aircraft at high ground speed (ore than 80 /s) in a ountainous area, significantly below the ridge line of the iediately surrounding terrain.

9 9 Architecture Scalar Tracking SR-DPE MR-DPE Basic Unit L channels K receivers 1 network Measured Quantities (Unknowns) ρ, ρ X = [x, y, z, cδt, x, y, z, cδt] X, α = [ϕ, θ, φ, ϕ, θ, φ ] # of Unknowns L K 2 K TABLE II C OMPARISON OF NUMBER OF UNKNOWNS AMONG DIFFERENT RECEIVER ARCHITECTURES Fig. 10. MR-DPE flight test. Clockwise fro top left: flight test aircraft, C-12C Huron; nose antenna (in front of cockpit canopy); tail antenna (on top of vertical stabilizer, not visible) and the easuring of antenna baselines with FaroAr; left-wing antenna. Fig. 11. Equipent rack (bright orange) as installed on board the flight test aircraft. Four USRPs are iediately visible in the botto-left corner. CSAC is hidden fro plain view. The TSPI tray (white, orange) can be seen on the right side of the cabin, ounted on the floor directly behind the co-pilot s seat. Fig. 13 shows the signal-to-noise ratio (SNR) history of one of the test points. Known officially as the Low-level Sidewinder transition [44], the test point saw the aircraft traveling below 1800 eters (above ean sea-level, MSL) in Kern River Valley, California, near Hockett Peak (2607 eters MSL) 1. Meanwhile, the surrounding terrain (to the east and west) is ore than 2000 eters within 1-ile radius, and 2400 eters within 2-ile radius. Note that there is a ajor dip in SNR N, W Fig. 12. Flight test experient setup. 1 of the 2 laptops is for data collection; the other is a back-up. Fig. 13. SNR history of the tail antenna of test point in Kern River Valley. Several satellites, as a group, are seen to experience dips in their SNR values due to terrain asking. for PRN #4, 14, 16 and 26, between t = 440 and t = 460. An exaination of the skyplot at the tie, shown in Fig. 14, reveals that these four satellites are concentrated on the East side of the aircraft and below 45-degree elevation. This spatial distribution of attenuated satellites corroborates our clai that the test point surely includes episodes when signal asking and ultipath are present. In contrast, Fig. 15 shows a baseline test point where the aircraft was traveling straight-and-level at 20,000 MSL, far away fro any possible terrain effect. The SNR during this test point is found to be tranquil.

10 10 (a) Kern River Valley Fig. 14. Skyplot at the tie of Fig. 13 (b) Steep-bank Turn Fig. 16. Scalar tracking vs. MR-DPE, during the two test points discussed in Section V. The traces of scalar tracking receivers are seen diverging on the right during both test points, due to loss of track. Fig. 15. SNR history of the tail antenna at high-altitude baseline test point A. Versus Scalar Tracking Fig. 16(a) shows the receiver outputs during the aforeentioned Kern River Valley test point. On the right is the colorcoded output of four scalar tracking receivers, each showing one receiver. Three out of the four scalar tracking receivers lost track shortly after entering the high-terrain area, north of Kernville, CA. In contrast, our MR-DPE ipleentation, shown on the right, aintained tracking throughout the test point. Fig. 16(b) is the result of a different data set collected at 20,000 cruising altitude. The aircraft was engaging in a 60- degree steep-bank left turn while the ground speed exceeds 370 /s. This aneuver deonstrates MR-DPE s ability to function noinally under high platfor-dynaics. On the left, the MR-DPE receiver aintained tracking of the GPS signals through out the 540-degree turn; on the right, the scalar tracking receivers lost track shortly after entering the turn. B. Versus Single-receiver DPE in this section we uses the sae Kern River Valley test point to copare the accuracy perforance between SR-DPE (with the four DPE receivers operating as standalone units) and MR-DPE (with the four receivers operating as a network). The error is defined as the ECEF coordinate difference, on the axis concerned, between the truth provided by the TSPI source and the said receiver architectures. Fig. 17(a) depicts the horizontal errors over the east-north plane. We can observe that MR-DPE provides a less noisy, ore consistent accuracy perforance throughout the 500- second long test point, and, for the ajority of the tie, is ore accurate than individual SR-DPE receivers. Fig. 17(b) shows the vertical errors during the sae period. We observe that MR-DPE provides higher vertical accuracy than individual SR-DPE receivers throughout the test point. This unequivocally deonstrates the superior robustness perforance of MR-DPE by aggregating the easureents fro ultiple receivers, which would have otherwise provided less accurate results if operated individually. VI. CONCLUSION In suary, we have presented a novel MR-DPE architecture, and provided highlight on the robustness advantage MR-DPE possesses over legacy scalar tracking and singlereceiver DPE architectures. We have conducted experients

11 11 (a) Horizontal error (East-north plane) This research is funded by Air Force Research Laboratory Sensors Directorate (AFRL/RY), Wright-Patterson AFB, OH, under contract FA The flight test experient was funded and executed under Project GRIFFIN, a Test Manageent Project of USAF TPS Class 16B. This report was prepared as an account of work sponsored by an agency of the United States Governent. Neither the United States Governent nor any agency thereof, nor any of their eployees, akes any warranty, express or iplied, or assues any legal liability or responsibility for the accuracy, copleteness, or usefulness of any inforation, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific coercial product, process, or service by trade nae, tradeark, anufacturer, or otherwise does not necessarily constitute or iply its endorseent, recoendation, or favoring by the United States Governent or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Governent or any agency thereof. R EFERENCES (b) Vertical error (Altitude) Fig. 17. Accuracy perforance during the Kern River Valley test point, as a coparison between the SR-DPE and the MR-DPE architectures. The truth was provided by TSPI. on a fixed-wing aircraft and have executed a wide range of flight profiles to fully test the ability of MR-DPE. We have deonstrated that MR-DPE provides iproved accuracy and robustness against ultipath and terrain asking, effects that are coonly found during critical flight phases. ACKNOWLEDGMENT The authors would like to express their heartfelt appreciation to the generous assistance they received fro the service and staff ebers of the United States Air Force Test Pilot School (USAF TPS), Edwards AFB, CA. Aong the are: Mr Jerey Cookson, Flt Lt Alexander Blackstock (RAF), Maj Joseph DeMonte, Maj Patrick Highland, Capt Mark Brodie, Capt John Wilder, Mr Lionel Banuelos, Ms Jennifer Harless, Mr Juan Garcia, Ms Debora Gonzales, Mr Krishtof Korda, and Mr Christopher Valdivia. The authors would also like to thank Mr Enyu Luo and Mr Derek Chen of University of Illinois for their support during experients. [1] Global Positioning Systes Directorate, Systes engineering and integration interface specification IS-GPS-200, [2] J. Beser and B. W. Parkinson, The application of NAVSTAR differential GPS in the civilian counity, Navigation, vol. 29, no. 2, pp , [Online]. Available: [3] A. King, Inertial navigation-forty years of evolution, GEC review, vol. 13, no. 3, pp , [4] D. Titterton and J. L. Weston, Strapdown Inertial Navigation Technology. IET, 2004, vol. 17. [5] P. Ward, The natural easureents of a GPS receiver, in Proceedings of the 51st Annual Meeting of the Institute of Navigation, 1995, pp [6] R. D. J. Van Nee, Spread-spectru code and carrier synchronization errors caused by ultipath and interference, IEEE Transactions on Aerospace and Electronic Systes, vol. 29, no. 4, pp , Oct [7] S. H. Kong, Statistical analysis of urban GPS ultipaths and pseudorange easureent errors, IEEE Transactions on Aerospace and Electronic Systes, vol. 47, no. 2, pp , Apr [8] G. X. Gao, M. Sgaini, M. Lu, and N. Kubo, Protecting GNSS receivers fro jaing and interference, Proceedings of the IEEE, vol. 104, no. 6, pp , Jun [9] R. L. Fante and J. J. Vaccaro, Evaluation and reduction of ultipathinduced bias on GPS tie-of-arrival, IEEE Transactions on Aerospace and Electronic Systes, vol. 39, no. 3, pp , Jul [10] W. Y. Ochieng, K. Sauer, D. Walsh, G. Brodin, S. Griffin, and M. Denney, GPS integrity and potential ipact on aviation safety, The Journal of Navigation, vol. 56, no. 01, pp , [11] X. Chen, F. Dovis, S. Peng, and Y. Morton, Coparative studies of GPS ultipath itigation ethods perforance, IEEE Transactions on Aerospace and Electronic Systes, vol. 49, no. 3, pp , Jul [12] J. J. Spilker, Vector delay lock loop processing of radiolocation transitter signals, US Patent 5,398,034, Mar. 14, [13] M. P. G. Lachapelle, Coparison of vector-based software receiver ipleentations with application to ultra-tight GPS/INS integration,, in Proceedings of the 19th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2006), Fort Worth, TX, Sep. 2006, pp [14] M. Lashley, D. M. Bevly, and J. Y. Hung, Perforance analysis of vector tracking algoriths for weak GPS signals in high dynaics, IEEE Journal of Selected Topics in Signal Processing, vol. 3, no. 4, pp , Aug [15] J. Liu, X. Cui, M. Lu, and Z. Feng, Vector tracking loops in GNSS receivers for dynaic weak signals, Journal of Systes Engineering and Electronics, vol. 24, no. 3, pp , 2013.

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Gao, Coputationally efficient direct position estiation via low duty-cycling, in Proceedings of the 29th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+ 2016), Portland, OR, Sep. 2016, pp [43] E. Wycoff and G. X. Gao, A Python software platfor for cooperatively tracking ultiple GPS receivers, in Proceedings of the 27th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+ 2014), Tapa, FL, Sep [44] Edwards AFB Flying and Airfield Operations (EDWARDSAFBI13-100), 412 OSS/OSAA, Oct Arthur Hsi-Ping Chu is a graduate student fro the Departent of Electrical and Coputer Engineering, University of Illinois at Urbana-Chapaign. He received his B.S. degree in electrical engineering fro National Taiwan University, Taipei, Taiwan in His research interests are in the developent and experient of GPS navigation algoriths. He is a student eber of IEEE and ION. Grace Xingxin Gao received her B.S. degree in echanical engineering and her M.S. degree in electrical engineering fro Tsinghua University, Beijing, China in 2001 and 2003, respectively. She received her PhD degree in electrical engineering fro Stanford University in Fro 2008 to 2012, she was a research associate at Stanford University. Since 2012, she has been an assistant professor in the Aerospace Engineering Departent at University of Illinois at Urbana-Chapaign. Her research interests are systes, signals, control, and robotics. She is a senior eber of IEEE and a eber of ION.