Sensitivity Analysis of 3D Building Modelassisted Snapshot Positioning

Transcription

1 Sensitivity Analysis of 3D Building Modelassisted Snapshot Positioning Rakesh Kuar and Mark G. Petovello Position, Location And Navigation (PLAN) Group Departent of Geoatics Engineering Schulich School of Engineering University of Calgary, Calgary, Canada BIOGRAPHIES Rakesh Kuar is a PhD candidate in the Position, Location And Navigation (PLAN) group of the Departent of Geoatics Engineering at the University of Calgary. He received his Bachelor of Engineering in Electronics & Counication Engineering and Master of Technology in Opto-Electronics fro Rajiv Gandhi Technical University, Bhopal India in 2002 and 2004, respectively. His research interests include iproving GNSS based navigation in Urban Canyon. Mark Petovello is a Professor in the Position, Location And Navigation (PLAN) group in the Departent of Geoatics Engineering at the University of Calgary. He has over 19 years of experience with various navigation research areas including software receiver developent, satellite-based navigation, inertial navigation, reliability analysis and dead-reckoning sensor integration. He is also registered as a professional engineer in the province of Alberta, Canada. ABSTRACT Global Navigation Satellite Systes (GNSS) has proven to be a viable and reliable solution in interference-free environent. However, in urban canyons reliable positioning is difficult to achieve in a cost-effective anner using standalone GNSS, due to ultipath probles and Non-Line-of-Sight (NLOS) signals. In this regard, the authors previously proposed a ethod whereby NLOS signals were used constructively by incorporating the inforation related to nearby reflectors with the help of a 3D building odel (3DBM). Since the proposed 3DBM-assisted algorith incorporates assistance data with the help of a 3DBM, the accuracy of the final estiated position will also depend on the accuracy of the assistance data and processing paraeters. This research investigates the sensitivity of the 3DBMassisted positioning algorith to various algorith paraeters and is tested using data collected in downtown Calgary, Canada. Results confir that errors in the 3DBM increase the error of the final position estiate. For given grid resolution, the size of the position grid does not affect the final solution. Increasing the grid resolution yields slightly better position estiates at the cost of uch higher processing load. The algorith is shown to be weakly sensitive to the accuracy of receiver s tie estiate; even with a tiing error of 10 s, the final estiated position was unchanged. Finally, the coherent integration tie (10 s vs 100 s) was shown to have very little ipact on position accuracy and deonstrates the capability of the algorith to provide good positioning accuracy without external data bit aiding even in dense urban environents. INTRODUCTION Satellite-based navigation and positioning has proven a viable solution in open sky. However, ultipath reains a doinant source of error, especially in urban environents where satellite obscuration (poor geoetry) and signal reflections (errors) fro nearby buildings conflate to produce poor perforance. Reflected or nonlight-of-sight (non-los, or NLOS) signals are delayed with respect to LOS signal; if the signal reaches the antenna via ultiple paths, the result is a ultipath effect that can be hard to detect in standard GNSS receivers. The fact that ultipath cannot be reoved by differential techniques liits positioning accuracy in ultipath prone areas such as urban canyons (Mishra & Enge 2011 and Ward et al 2006). Furtherore, itigation of ultipath errors becoes difficult when the Doppler shift between the direct and secondary propagation paths is very sall (approxiately 1-2 Hz), which is usually the case in urban canyons where the reflectors are often parallel to the direction of otion of the receiver. Aple research has been done towards ultipath itigation in order to iprove GNSS-based navigation in urban environents. Most of the research has been directed towards identifying and eliinating NLOS signals. These ethodologies can be broadly categorized as: ION GNSS+ 2016, Session B6, Portland, OR, Septeber 2016 Page 1 of 11

2 Traditional ethods of GNSS integration with Inertial Navigation Syste (INS), where the copleentary benefits of both the systes are utilized (Soloviev & Graas 2009, Petovello et al 2008 and Petovello & Lachapelle 2006 etc.) Modified receiver processing strategies, where the receiver architecture is odified for better signal processing aspects (Xie & Petovello 2014 and Van Graas et al 2009) Using ulti-constellation GNSS for increasing the nuber of easureents (Groves et al 2012 and O Driscoll et al 2010 etc ) and paraeter estiation techniques (He & Petovello 2013 and Van Nee 1992) Apart fro ethods entioned above, 3DBMs have been used for identification of NLOS signals towards a constructive use of NLOS signal. At high level, 3DBMs are digital representations of cities which contain the relevant geospatial inforation (Frere 1998). Although 3D odels have been used for non-gnss applications for decades; in urban areas and indoors they have found GNSS related applications, only in the recent past, as described below. Bradbury et al (2007) used a 3D city odel for deterining satellite availability and analysing signal degradation in ultipath environent, based on siulated scenarios. Ben-Moshe et al (2011) have used a 3D odel for iproving perforance in signal degraded areas and also in their work they have tried to use the present position inforation to help in building the surroundings. Suzuki and Kubo (2012) used 3D city odels for positioning in urban canyons using ultipath siulations. Bourdeau (2012) has shown that a 3D odel of the environent can be used to predict the geoetric paths of NLOS signals using 3D city odel. To detect and predict ultipath situations in urban areas without any additional hardware a 3D representation of surrounding was proposed by Obst et al (2012). More recently, 3D city odels have been used for: a portfolio approach-based positioning in urban canyons (Groves et al 2013); ultipath and NLOS delay estiation based on particle filter (Suzuki and Kubo 2013); unique usage of 3DBM for skyline positioning ethod (Petovello and He 2016) and for ultipath paraeter estiation (Kuar and Petovello 2015). Gu et al (2015) have used 3DBM along with onboard sensors for self-localization of a vehicle in urban canyons. Adjrd and Groves (2015) have used real-tie etre-level obile positioning in outdoor urban environents by aking use of spatial data in the for of 3D city odels. Hsu et al (2016, 2016a) have used 3D building odel, for pedestrian positioning using GPS/GLONASS/QZSS in urban canyons and for autonoous driving position applying differential GNSS. Augenting GNSS syste with additional inforation fro 3DBMs has proven to be a viable approach for iproved accuracy in urban canyons; either by using shadow atching algoriths (such as: Groves 2011, Wang et al (2013, 2015), Yozevitch and Ben Moshe 2015) or by using path delay of NLOS signals constructively (Kuar & Petovello 2014 and Bourdeau 2012). Furtherore, Kuar & Petovello (2016) have shown that constructive use of NLOS signals can be used for snapshot-based positioning (using up to a few hundreds of illiseconds of GNSS signal saples), although that work assued the receiver had accurate knowledge of tie and only considered a fixed set of processing paraeters. This work was unique since the ethod was based on providing position solution with a snapshot of GNSS data, without filtering. Moreover, the position estiate was based on a unique atching algorith that differs fro traditional pseudorange-based positioning algoriths. However, as explained in following section, this positioning ethod depends on various algorith paraeters that, in principle, can affect the accuracy and perforance of the positioning ethod. Hence the sensitivity analysis of the positioning algorith with respect to key algorith paraeters needed to be analyzed. The objective of this paper is to expand upon the authors previous work (Kuar & Petovello 2016) in order to better understand the sensitivities of the algorith to various paraeters such as length of the position grid, resolution of the position grid, and error in estiated receiver tie, error in 3DBM and effect of different coherent integration ties. METHODOLOGY The 3DBM-assisted snapshot positioning concept is depicted and suarized in Figure 1; for ore detailed inforation refer to Kuar & Petovello (2016). The approach first defines a Position Grid (PG) around the user s approxiate position. Each point in the position grid is called a Candidate Point (CP), one of which will be selected as the final position estiate. For each CP, the 3DBM and a ray-tracer are used to predict the nuber of signal paths and additional delay of each NLOS signal relative to LOS signals ( path delay ). This inforation is then used to fit 1, 2 or 3-path signal odels to the receiver s correlator outputs using least-squares (LSQ) estiation. Intuitively, the LSQ fit will be best as easured by the least-squares residuals for the CP that ost closely atches the user s true position and this CP is selected as the final position estiate. The LSQ fit in Figure 1 is perfored at all CPs and for each satellite and the root-su-of-squared (RSS) of residuals are coputed in each case. For a given position, the RSS of residuals across all satellites are sued together at each CP and the CP with sallest total RSS of residuals value is selected as the final position solution. 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3 Figure 1: Top level ethodology of a 3DBM-assisted snapshot positioning using all NLOS signals constructively Using this approach, Kuar and Petovello (2016) showed that the 3DBM-assisted snapshot positioning algorith provided an RMS error iproveent of 64% across-track and 48% in horizontal plane copared to pseudorangebased receivers when used in downtown Calgary, Canada. More interestingly, the results indicated a relationship between error and sky-visibility; better perforance in poor sky-visibility. However, these results were only obtained for a fixed set of algorith paraeters, naely: a PG size of 30x30, a grid resolution of 1, correct receiver tie (no coarse-tie estiation) and coherent integration tie of 100 s. In order to provide a solid support to above results it is iportant to analyze the results for the sae data set with different sets of processing paraeters. More precisely, the algorith needs to be tested for sensitivity towards: a) Errors in approxiate user position (and thus the size of the PG): Since the algorith s approach is based on initial approxiate user position, the uncertainty in this position would affect the size of position grid. For a larger uncertainty, the size of grid should be large enough to capture the true location. Hence the sensitivity analysis of the algorith with respect to size of positon grid becoes essential. b) Resolution of the PG: As entioned above the final estiated position is one of the candidate points inside the position grid, as deterined fro the output of the atching algorith. That eans the resolution of the grid (distance between two nearest candidate points) will affect the accuracy of the final estiated positon. Thus, the sensitivity analysis of the algorith with respect to resolution of the grid needs to be done. c) Tiing error: The proposed approach uses snapshot positioning algorith where receiver ties are not often known precisely, leading to errors in satellite positions due to incorrect transit tie (i.e., the coarse tie proble). The sensitivity of the proposed algorith to receiver tiing errors is therefore critical to understanding its practicality for real-world application. d) Errors in the 3D building odel: As entioned earlier, the nuber of signal paths and NLOS signal delays are predicted using a 3DBM and ray-tracing, hence any error in 3DBM would affect the quality of predicted paraeters and thus the final estiated position. The effect of error in 3DBM on the algorith s perforance is therefore analyzed here. For sake of siplicity, only the biases in the 3DBM are considered; rando errors (e.g., between buildings) are not considered. e) Coherent integration tie: The power of correlator output depends on coherent integration tie. Since these correlators are used for atching with predicted paraeters, the coherent integration tie will affect the quality of atch and, in turn, the quality of final estiated position. Hence, the sensitivity analysis of the algorith with respect to coherent integration becoes iportant. Considering these factors, this paper s focus is towards accessing the sensitivity of the 3DBM-assisted snapshot positioning to the size and resolution of the PG, the effect of transit tie, 3DBM errors and coherent integration tie: - Effect of PG size are assessed directly by using different sizes of PG and coparing perforance in ters of horizontal position s RMS error - The grid resolution s effect is assessed in a siilar anner and its effect is considered in ters of processing load and accuracy of the final position estiate - Sensitivity to tiing error is first assessed theoretically and then tested by injecting an intentional tiing error and coparing results corresponding to the true tie - Effect of a translation in the 3DBM coordinates are assessed by adding errors of different agnitudes to the vertices coordinates of the 3DBM and coparing results - Sensitivity to coherent integration tie is assessed by coparing the RMS error of horizontal position for 100 s, 20 s and 10 s coherent integration ties The following section details about the data collection, data processing and results. ION GNSS+ 2016, Session B6, Portland, OR, Septeber 2016 Page 3 of 11

4 DATA COLLECTION AND PROCESSING In order to analyse the sensitivity of the 3DBM-assisted snapshot positioning algorith, live data was collected in downtown Calgary. The data collection setup consisted of a NovAtel SPAN-LCI reference syste, a front-end for interediate frequency (IF) saple collection, a NovAtel antenna and a base station. The reference trajectory was obtained with NovAtel s Inertial-Explorer software using a tightly coupled forward and backward soothing configuration along with differential corrections fro the base station. All of the equipent were ounted on a test van (except for the base station) and the vehicle was driven through downtown Calgary with a axiu speed of about 15 /s. The IF data was processed using the University of Calgary s GSNRx software receiver (Petovello et al 2009) to generate the correlators outputs for input into the LSQ algorith. The default processing paraeters are shown in Table 1 (these will vary depending on the paraeters being evaluated). Table 1: Default Processing Paraeters Search Space Search Step Integration Tie Position solution rate PG size PG resolution Doppler Doain Code Phase Doain Doppler Doain Code Phase Doain 100 s 1 Hz 30x30 1 ±150 Hz ±300 2 Hz 10 Figure 2 shows the trajectory, which is approxiately 2.5 k in length. Although the analysis considers ultiple points along the trajectory, a detailed analysis is perfored at the location indicated by the yellow star in Figure 2. The selected point has sky visibility of approxiately 55% and has reflectors to the north and south and is otherwise representative of any other points in the data set. EXPERIMENTAL RESULTS AND ANALYSIS This section provides results and assessent of the sensitivity of the 3DBM-assisted snapshot positioning to different paraeters. Sensitivity To The Size of Position Grid Figure 3 shows the RSS of residuals across a 30x30 position grid for PRN 1 at the location under consideration. In this case, the true position is located at the centre of the grid and, as expected, the residuals are sallest around this point. The RSS of residuals for the sae PRN for a 200x200 grid is shown in Figure 4, where the shaded regions depict the surrounding buildings (all CPs falling inside a building are excluded fro the position deterination). Ideally, it is expected that the RSS of residuals should increase for CPs farther fro the true location. In general, this is the case in Figures 3 and 4, especially in the northsouth direction (although the values appear to be approxiately binary in nature). However, in the east-west direction the 200x200 grid shows several ore CPs with low RSS of residuals east of the true location. Generally the position is not coputed fro a single satellite; hence this is not a ajor concern. Nevertheless, if only this particular satellite is used, the coputed position has an error of 20 in the east and 5 in the south. This also shows how the proposed algorith can be used to deterine location using only one satellite. The error in case of 200x200 position grid apparently sees to be larger ; however, it is worth entioning that the resolution of the position grid is 5 in this case as opposed to 1 in case of 30x30 grid. Figure 2: Reference trajectory and epoch considered for detailed sensitivity analysis. The user at the epoch considered for the sensitivity analysis is highlighted in yellow star (towards west side) on the trajectory. Figure 3: RSS of residuals for PRN 1 (Az: 297, Elve: 42 ); size of position grid is 30x30 etres ION GNSS+ 2016, Session B6, Portland, OR, Septeber 2016 Page 4 of 11

5 Figure 4: RSS of residuals for PRN 1(Az: 297, Elev: 42 ); size of position grid 200x200 etres; shaded regions are buildings and are excluded fro position grid for RSS of residuals calculation Figure 6: RSS of residuals due to all PRNs present at this epoch; size of position grid is 200x200 etres; shaded regions are buildings and are excluded fro position grid for RSS of residuals calculation. The white star represents the final estiated position. To explain the results in Figure 4, we consider that PRN 1 has an aziuth of 297 and so its signals will ostly likely reflect fro building south and/or east of the user. In this case, since there are no significant reflectors to the east, the signal is ost likely being reflected fro a building towards south of the user. However, for a reflector whose noral vector points towards north, a change in the eastwest position of the user will not have uch effect on the predicted path length (this is indirectly explained by equation (1) below) and hence the LSQ RSS of residuals should be nearly constant eaning the algorith is not as sensitive in this direction. Figures 5 and 6 show the RSS values sued across all satellites for the 30x30 grid and the 200x200, respectively. Coparing these figures with the results in Figures 3 and 4 (PRN 1 only), adding inforation fro ore satellites gives fewer regions with sall RSS residuals, which is expected. In this case, the final position estiate in the 200x200 case has an error of 5 toward the west and 5 toward the south. The iproveent in east-west error (with respect to using only PRN 1) is obvious due to the inclusion of RSS of residuals fro other satellites. Results presented here indicate that increasing the size of the position grid does not change the final estiated position. The process of estiating the final position for an increased grid size of 200x200 was repeated for the entire trajectory using approxiately 10 inutes of data, and coputing a solution every 10 seconds. This was done in order to avoid having repeated geoetry when the vehicle was stopped and also to avoid huge processing load. The error statistics is shown in Table 2. Table 2: Coparison of RMS of horizontal error for entire trajectory using different grid sizes Horizontal Error for 30x30 grid and grid resolution 5 Horizontal Error for 200x200 grid and grid resolution Figure 5: RSS of residuals due to all PRNs present at this epoch; size of position grid is 30x30 etres. The white star represents the final estiated position. Results in Table 2 show the RMS error is approxiately sae for two different grid sizes. These results indicate ION GNSS+ 2016, Session B6, Portland, OR, Septeber 2016 Page 5 of 11

6 that the algorith is arginally sensitive to the size of position grid. Sensitivity To Resolution of Position Grid The resolution of the position grid, or grid resolution, is an iportant factor in ters of obtaining the final position estiate. Increasing the grid resolution results in increased nuber of candidate points for a given grid size. This, in turn, should increase the likelihood of obtaining a candidate point closer to the true location but the processing load will have to increase to accoodate the increased nuber of candidate points under consideration. In order to analyze the effect of the grid resolution on the final estiated position accuracy and on the processing load, different grid resolution of 0.5, 1 and 5 were considered for a position grid of 30x30. The 2D error in final estiated position and nuber of Candidate Points for a grid resolution of 0.5, 1 and 5 are presented in Table 3. As expected, the horizontal error (2D error) for the finer grid resolution is better than for the larger grid resolution. Also shown in Table 3, this coes with a draatic increase in processing load, as reflected by the nuber of candidate points. Table 3: Perforance coparison for different grid resolutions for a 30x30 grid Grid Resolution (GR) Nuber of CPs change in CPs (with respect to 1 GR) N/A 2D error 5.7 Relative Error (with respect to 1 GR) N/A These results are as expected, since the increase in grid resolution is expected to give better positioning accuracy at the cost of increased processing load and vice-versa. Based solely on these results, it would see that a 0.5 resolution is not worth the coputational tradeoff but a 1 or 5 resolution would be sufficient, depending on the accuracy requireents. These results also suggest that the 3DBM-assisted snapshot positioning algorith can be ade ore efficient by ipleenting a ulti-stage approach. More precisely, a coarser gird can be used to obtain a refined position estiate (copared to the approxiate position used to initialize the algorith); then a saller, finer grid could be generated around that point to get the final estiate; this is left as future work. Sensitivity to Error in Transit Tie In traditional pseudorange-based positioning, because of satellite otion (~4 k/s), an error of 1 s corresponds to an error in range rate of up to about 800 ; which eventually leads to hundreds of etres of error in the position estiate (Van Diggelen 2009). Since proposed ethod is based on positioning with a snapshot of data, eaning that precise tiing inforation cannot be extracted, the effect of receiver tiing error becoes a crucial part of analysis. In order to analyze the sensitivity of the 3DBM-assisted positioning algorith with respect to the tiing error, Figure 7 shows a satellite at location a at a particular tie, along with the signal path that is incident on a reflector located near the receiver. The reflector is at a perpendicular distance d fro the reflector and θ is the angle of incidence defined with respect to the noral vector of the reflecting surface. Although Figure 7 is shown in 2D, the 3D scenario is effectively the sae but is also a function of aziuth % (ore) % (better) % (less) % (poor) Copared to the case of a 1 grid spacing: - The horizontal error for the 0.5 grid resolution is only slightly better (reduction of 0.4 copared to the case of 1 grid spacing) but at the cost of a nearly three-fold increase in the nuber of CPs, and hence processing load. - The horizontal error for the 5 spacing increased by 1.4 (25% increase) with a 95% reduction in processing load. Figure 7: Scenario with a reflector near-by receiver and reflections reaching the receiver ION GNSS+ 2016, Session B6, Portland, OR, Septeber 2016 Page 6 of 11

7 For this given scenario the path delay (D) of the reflected signal, is given by D = 2*d*cos(θ) (1) Any error in satellite transit tie (as derived fro the receiver tie), would lead to a wrong satellite position. Hence, assuing an error in transit tie, the satellite would be at a different location along the orbit, say at b in Figure 7. Taking the tie derivative of the above equation yields dd dθ = -2*d*sin(θ) * (2) dt dt where, ( dθ/ dt ) is the rate of the incident angle. An approxiate worst-case (largest) value for this is the angular rate of the satellites as viewed fro the centre of the Earth (i.e., the satellite s ean anoaly): Table 4: Error in path delay, aziuth and elevation of point of intersection of incident ray and reflector as a function of transit tie error Error in Transit tie 0 s (true transit tie) Aziuth to point of intersection Elevation to point of intersection s s s Path delay Error in delta path delay NA V s ax( dθ/ dt) (3) R where V is the satellite orbit speed and R is the distance s fro the centre of the Earth to the satellite. Continuing with the pessiistic analysis, d = 0.5 code chips and θ=90 (such that sin(θ) = 1); this would yield a axiu path delay of 1 chip which is the largest that can ipact the correlation function peak. Then, using GPS specific values for V s and R, the worst-case error in path delay is less than 5 c per second of tiing error. Given the level of code-phase noise and the agnitude of orbit and atosphere errors, this level of error is quite sall. It is also worth noting that the value above (5 c per second of tiing error) is uch saller than the absolute code phase error that is used for traditional coarse-tie positioning approaches (Van Diggelen 2009), which are proportional to the geoetric range rate between the receiver and satellite (up to about 900 /s). In this regard, the 3DBM-assisted snapshot positioning approach should be uch better suited to coarse-tie positioning applications. To confir the above, Table 4 shows the error in path delay coputed for PRN 1 for tiing errors up to 10 s; as can be seen, the path delay errors are 2 c or less. These values are less than the worst case error analysis perfored above, which is 50 c per 10 second tiing error (as explained above). In order to show the effect of transit tie error on the 3DBM-assisted snapshot positioning algorith, Figure 8 shows the difference in RSS of residuals for PRN 1 copared to what was shown in Figure 3, after introducing an error of 10 s in the transit tie. The difference in RSS of residuals is nearly zero everywhere. Figure 8: Difference in RSS of residuals for PRN 1 (Az: 297⁰, Ele: 42⁰), relative to Figure 3, due to a transit tie error of 10 s The difference sees larger at one of the CP (difference of ore than 0.9) because of a change in the nuber of paths at that CP due to tiing error of 10 s (copared to the true tie case). Since predicting the wrong nuber ION GNSS+ 2016, Session B6, Portland, OR, Septeber 2016 Page 7 of 11

8 of paths would lead to larger RSS of residuals copared to that corresponding to true nuber of paths, these results ake sense. More iportantly, the final estiated position was at sae location (sae CP) for a grid resolution of 1. By extension, in worst case scenarios where the tiing error results in wrong nuber of paths for ost of CPs, the 3DBM-assisted snapshot positioning algorith is expected to provide an estiated position corresponding to a CP with a valid and least RSS of residuals, taking care of over-fitting and under-fitting of LSQ (details in Kuar & Petovello 2016). Hence, even with 10 s error in transit tie, the error in estiated position is significantly saller than that observed in traditional coarse tie approaches where errors can be as large as 800 per second of tiing error. In other words, using path delay for position coputation is nearly insensitive to the transit tie error. The process of tiing error injection was repeated for the entire trajectory of 2.5 k for an error injection of 5 s. It was observed that the ean error in path delay was 0.08 etres which eventually ended up in sae candidate point as final estiated position. Sensitivity To Error in 3DBM Accuracy of 3DBM is an iportant factor for 3DBMassisted snapshot positioning ethod. In order to analyze the effect of the error on the final estiated position, bias of 1 and 3 were injected in the vertices coordinates of the given 3DBM. In order to analyze the effect of errors in 3DBM along and across the trajectory, errors of agnitude 1 and 3 were injected in east direction (along-track) and north direction (across-track) respectively, at the epoch under consideration (yellow star in Figure 2). Table 5 copares the 2D positioning accuracy due to errors of different agnitude in the 3DBM. The 2D position error corresponding to the case with no injected error is considered as reference for coparison purposes. Based on the results fro Table 5 and copared to the reference (no error injection) case: - Error up to 3 along track direction has no effect on the 2D position error. - 2D position error reains the sae (as copared to no injected error in 3DBM) for an error of 1 across track. - The 2D position error increases by 58% approxiately, for an error injection of 3 across track in the 3DBM. Table 5: Effect of 3DBM error on estiated position for a grid size of 30x30 and resolution of 5 3DBM Error 2D RMS Error () Change in 2D RMS Error due to biases N/A 1 along track 7.1 No change 1 across track 7.1 No change 3 along track 7.1 No change 3 across track % These results are as expected, since the effect of across track error (injecting error in north coordinate of the vertices) in 3DBM would be ore profound; this fact is indirectly explained by equation (1) above. Since the epoch considered (yellow star in Figure 2) lies on the trajectory along east-west directions and buildings (possible reflectors) are present in north-south direction, error injection (in vertices coordinates of the 3DBM) in north-south direction (across track) would result in an obvious position error (as suggested by equation (1)). Since the grid resolution considered here is 5, the effect of across track error of 1 is effectively absorbed by the grid resolution. By extension, the effect of across track error in 3DBM would be visible as an additional error in the final estiated position, for finer grid resolution. Sensitivity To Coherent Integration Tie The sensitivity of the 3DBM-assisted snapshot positioning algorith with respect to the coherent integration tie (CIT) was analyzed for CIT of 10 s, 20 s and 100 s. The resulting RMS horizontal position errors along with ean and standard deviation for these cases are shown in Table 6. Table 6: Effect of coherent integration tie on 3DBMassisted snapshot positioning algorith Coherent Integration Tie Mean Std. Deviation RMS Horizontal Position Error 100 s s s ION GNSS+ 2016, Session B6, Portland, OR, Septeber 2016 Page 8 of 11

9 Copared with the 100 s case, the RMS of horizontal error for 20 s CIT differed by only 20 c. There is arginal degradation of positioning perforance (RMS error increases by 10 %) when CIT was reduced to 10 s fro 100 s. This arginal degradation in case of 10 s CIT-based correlators can be attributed to noisier correlator output in case of 10 s CIT, as can be seen fro the statistics in Table 6 and noralized correlator plots in Figure 9. Figure 9: Noralized correlator outputs for 10 s and 100 s coherent integration ties. The correlator output corresponding to 10 s is noisier copared to that of 100 s. Hence based on the results the perforance of the 3DBMassisted snapshot positioning ethod is considered arginally sensitive to CIT. The result further suggests that external data bit aiding would be unnecessary while using 3DBM-asisted snapshot positioning ethod; which in turn iplies a siplified architecture of receiver for positioning in dense urban areas. SUMMARY AND CONCLUSIONS This paper provides a sensitivity analysis of 3DBMasisted positioning ethod with respect to size of the position grid, resolution of the position grid, tiing errors, error in 3DBM and coherent integration ties. The sensitivity of 3DBM-assisted positioning algorith to the size of the PG was deterined by coparing results for a 30x30 with those of a 200x200. It was observed that the effect of size of position grid on the algorith was entirely based on geoetry of the reflectors around the receiver position, as expected. It was also observed that the algorith s perforance in ters of final estiated position, does not change with the change in size of the position grid; i.e., 3DBM-assisted snapshot positioning ethod is nearly insensitive to size of positon grid. The perforance of the algorith in ters of accuracy and processing load was observed to be dependent on the grid resolution. It was observed that erely increasing the resolution to 50 c fro 1 resulted in a 287% increase in processing load with a arginal iproveent in accuracy of 7% (0.4 ). On the other hand, decreasing the grid resolution to 5, the processing load was reduced by 95% at the cost of 1.4 ore error (25% increase). The algorith s sensitivity in ters of positioning accuracy with respect to tiing error was analyzed for transit error up to 10 s. Results indicated that even with an error of 10 s in transit tie, the error in estiated position was less than a etre. Hence it can be concluded that the 3DBM-assisted snapshot positioning algorith is nearly insensitive to tiing error. Hence, the 3DBMassisted positioning technique proposed by sae authors (Kuar & Petovello 2016), can be used for low cost urban navigation, for exaple, in autootive applications and in weak signal environents, with a snapshot of GNSS data without tracking. The algorith s sensitivity with respect to error in 3DBM was analyzed for error of 1 and 3, along and across track. The analysis was done by coparing the 2D position accuracy of the algorith, with and without additional (injected) error in 3DBM. It was perceived that the effect of error in 3DBM, across track, was ore profound; an increase in 2D position error of 57 % was observed due to an additional error of 3 across track. Hence, the algorith sees ore sensitive to 3DBM error s coponent in across track and less sensitive to the error coponent towards along track. Finally, the algorith s sensitivity with respect to coherent integration tie was analyzed. The RMS error of horizontal position was copared for 100 s, 20 s and 10 s coherent integration ties. The arginal iproveent of positioning perforance by 10% for 100 s coherent integration copared to 10 s coherent integration, suggests external data bit aiding is unnecessary even while navigating in dense urban areas and hence siplifies receiver architecture. ACKNOWLEDGMENTS Alberta Innovates Technology Futures (AITF) is acknowledged for supporting the first author via an AITF Graduate Student Scholarship. ION GNSS+ 2016, Session B6, Portland, OR, Septeber 2016 Page 9 of 11

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