auv terrain relative navigation using coarse...
TRANSCRIPT
AUV TERRAIN RELATIVE NAVIGATION USING COARSE MAPS
Deborah K. Meduna1
[email protected] M. Rock1,2
[email protected] S. McEwen2
1 Dept. of Aeronautics and AstronauticsStanford University, 496 Lomita Mall
Stanford, CA 94305
2 Monterey Bay Aquarium Research Institute7700 Sandholdt Road
Moss Landing, CA 95039
Abstract
Terrain Relative Navigation (TRN) provides an en-abling technology for drift-free, low-cost navigationon many existing underwater vehicles. While high-performance TRN systems have been demonstrated us-ing high-quality sensors and terrain maps, these systemscan be very costly and the high-resolution bathymetrymaps used are of limited availability. To extend theapplicability of TRN, the authors previously addressedthe performance trades in utilizing lower-cost, lower-accuracy sensor systems for terrain navigation. The cur-rent paper addresses the additional extension of TRN tocoarse, ship-based maps which are more widely avail-able. Performance studies assessing TRN performanceover varying map resolution are presented for two sets ofreal-time field trials, performed over different terrains inMonterey Bay, CA. A posterior Cramer-Rao bound anal-ysis provides insight into the terrain characteristics mostrelevant to the observed TRN performance reductionover coarser maps. Finally, the results from these studiesare validated against 20m ship-based bathymetry maps.In these studies, moving from a 1m to a 30m resolu-tion map is observed to reduce TRN estimation accuracyby a factor of only 2-3, resulting in sub-map resolutionaccuracies. The presented results indicate that, for therange of resolutions considered, coarse ship-based mapscan potentially be utilized in place of higher-resolutionAUV-based maps with only mild performance losses.
1 Introduction
Terrain relative navigation can enable drift-free, low-cost navigation for underwater vehicles, providing apowerful alternative to other underwater navigationmethods including periodic resurfacing for GPS, dead-
reckoning with high-accuracy inertial sensors, and/ordeploying transponder arrays. TRN generates vehi-cle position estimates by matching terrain measure-ments against a stored terrain map. The technologywas initially developed with TERCOM (Terrain Con-tour Matching) for cruise missile navigation, consistingof batch correlation of altimeter measurements [1].
Since TERCOM, numerous variants of TRN havebeen successfully developed for a range of platforms, in-cluding underwater vehicles [2], [3], [4], [5], [6], [7],[8], [9]. Many of these implementations report highaccuracy performance by using a combination of high-quality instrumentation and high-accuracy Digital Ele-vation Model (DEM) terrain maps. Nygren, for exam-ple, demonstrated high performance through TERCOM-style batch correlation of large sonar measurementpatches, created from combining several multibeamsonar pings with well-known relative spacing [3].
In [8], the authors demonstrated meter-level accu-racy in field trials on the Monterey Bay Aquarium Re-search Institute (MBARI) mapping AUV, using either anAltimeter or Doppler Velocity Log (DVL) sonar mea-surements for correlations against a high-resolution (1meter) DEM map. The presented results were aimedat evaluating the affect on TRN performance of usinglower-cost, lower-accuracy vehicle instrumentation. En-abling TRN on these lower-cost sensor systems signifi-cantly expands the range of potential TRN applications.
This paper continues this work by addressing the ad-ditional concern of the performance effects in transition-ing to coarser resolution terrain maps. Enabling effec-tive TRN over low-accuracy terrain maps can substan-tially increase its applicability. For example, while DEMmaps are available for much of Monterey Bay, Califor-nia, only a small percentage have been generated withmeter-level accuracy (e.g. by MBARI’s mapping AUV).The vast majority of available DEM maps are derived
from ship-based bathymetry, resulting in resolutions onthe order of 10’s of meters.
Utilizing coarser resolution maps for terrain naviga-tion, however, introduces important performance tradeoffs that have not been thoroughly addressed in AUVnavigation literature [10]. The achievable accuracy ofa TRN system is expected to be inversely related to thecoarseness of the utilized terrain map. However, the rateof performance degradation and its dependence on un-derlying terrain properties is not well understood.
This paper explores the performance trades involvedin moving to coarse DEM maps for TRN through casestudies on two sets of real-time field trials, performedover different terrains in Monterey Bay, CA. A TRNconfiguration is presented which utilizes DVL sonarfor terrain correlation and a point mass filter for esti-mate propagation. Empirical studies are presented usingDEMs created at varying resolutions to assess TRN per-formance over coarser maps.
Reduced map resolution is observed to effect TRNperformance through both decreased accuracy of themodeled terrain elevations, and a decreased ability ofthe map to represent the true terrain gradient variability.An analytical performance study based on the posteriorCramer-Rao bound is presented which provides a meansfor quantifying these effects for a given terrain. Finally,the results of these studies are validated against TRNperformance with 20m ship-based bathymetry maps.The presented investigation shows that the TRN perfor-mance degrades gradually with decreased map resolu-tion, implying that coarse ship-based maps on the orderof 10s of meters can be effectively used for TRN withoutsignificant performance losses.
2 TRN Filter Implementation
The performance analysis and field trials presented inthis paper are generated using a point mass filter TRNalgorithm similar to that described in [8]. The terrainnavigation system is modeled by the equation pair:
xt = xt−1 + ut + rt (1)yt = h(xt) + et (2)
with vehicle state given by:
xt = [xNortht xEast
t ]T . (3)
Equation 1 models the state propagation, where ut isthe change in vehicle state measured by an inertial nav-igation unit, and rt ∼ N(0,Σr) is the process noisedetermined by the inertial drift rate and distance trav-eled. The sonar measurement is modeled by Equation
2, where yt ∈ RN is the N projected sonar measure-ments of terrain depth, h(xt) is the corresponding ter-rain depth at the sonar projected beam locations relativeto state xt, and et ∼ N(0,Σe) is the sonar measurementnoise, modeled as 4% of the measured range.
Unfortunately, the true terrain surface in the measure-ment equation, h(xt), is not generally known. Instead,the terrain surface must be approximated by a model.In the geoscience community, the most common suchterrain model is the DEM. A typical DEM representsthe terrain by a grid of elevation values, uniformly dis-tributed in North and East. The resulting model for ter-rain depth at any given location, h(xt), is given by Equa-tion 4:
h(xt) = h(xt) + vt + δz (4)
where h(xt) is the DEM estimated terrain depth at lo-cation xt, vt is the associated DEM modeling error, andδz is a depth bias term which accounts for variation intide levels between the time of the DEM creation and thecurrent vehicle mission.
In order to estimate terrain depth at any inter-grid lo-cation, a linear interpolation method is utilized:
hi(xt) =n∑
j=1
λjhMi,j [xt] (5)
where hMi,j [xt] is the DEM value at the jth grid cell
in the neighborhood of measurement yt,i, and λj isthe associated interpolation weight. Many interpolationmethods can be represented as in Equation 5, includingnearest-neighbor, bilinear, and bicubic. Each of thesewere evaluated for the current filter implementation andit was found that while bicubic interpolation performedthe best, bilinear interpolation provided a better perfor-mance versus speed trade-off and was thus selected forthe current filter. The tendency of higher order interpola-tion methods to give small performance gains at the ex-pense of large computational effort has also been notedby several in the geoscience community [11], [12].
Using this interpolation method for terrain depth esti-mation results in the new measurement equation,
yt,i = λThMi [xt] + δz + vt,i + et,i. (6)
For simplicity, the DEM modeling error hasbeen approximated as a zero-mean guassian, vt ∼N(0, diag(σ2
v)), where the covariance is given by:
σ2v,i = λT Cov(hM , (hM )T )λ (7)
= λTσ2(hM ) + λT[A− γ(hM , (hM )T )
]λ
where Aij = 0.5(hMi − hM
j )2, and γij is the variogramof the distance between grid points hM
i and hMj . A var-
iogram describes the expected mean square difference
between points at a specified spatial separation, and arefundamental to the field of geostatistics, initially devel-oped by [13]. They are commonly used as a convenientmeans of inferring spatial covariance [14]. In the currentimplementation, a fractal variogram model is utilized,given by Equation 8:
γ(s) = 0.5E[((h(x)− h(x+ s))2]= a||s||2(3−D) (8)
where s is a separation (lag) distance between datapoints, and D is the fractal dimension, related to ter-rain roughness. Fractal models, initially developed by[15], are commonly used for representing variability innatural terrains.
Assuming that the sonar measurement noise is uncor-related with the map error and errors between beamsare independent, the probability of acquiring the currentsonar measurement, yt, given vehicle state xt and depthbias δz is given by:
p(yt|xt, δz) = α exp
(−1
2
N∑i=0
wi
[yi,t− hi(xt)− δz
]2)(9)
where
wi =1
σ2e,i + σ2
v,i
. (10)
The resulting likelihood distribution is thus a three-dimensional function over xt and δz. However, thedepth bias term, which represents the depth offset dueto tidal variations, is simply the offset between the mapdepth and the true vehicle depth plus altitude. As a re-sult, δz is deterministic given the true vehicle position.Thus, the three-dimensional probability in Equation 9can be approximated as a two-dimensional surface byreplacing the depth bias term with its maximum likeli-hood estimate, given in Equation 11. This approxima-tion is exact for a single time step.
δz(xt) =1∑i wi
∑i
wi(yi,t − hi(xt))
= ηwT (yt − h(xt)) (11)
The effect of this resulting depth bias term is to con-strain the search to the sonar-measured altitude abovethe map and apply contour matching rather than depthprofile matching. Both [3] and [4] use a similar contour-matching based TRN algorithm. The resulting two-dimensional likelihood of making measurement yt at lo-
cation xt is given by Equation 12:
p(yt|xt) = α exp
(−1
2
N∑i=0
wi
[Π(yt − h(xt))
]2i
)(12)
whereΠ = IN − ηuwT . (13)
The likelihood surface is clearly highly nonlinearin the state, primarily due to the natural nonlinearityof the terrain function, h(x). This nonlinearity hasbeen widely noted for terrain navigation filter applica-tions, and increases with fewer measurement beams.As a result, a full Bayesian, non-parametric filter isneeded for successful tracking of the vehicle state es-timate. In this work, a point mass filter (PMF), or his-togram filter, is utilized for state propagation. For low-dimensional search, the PMF is advantageous over othernon-parametric filters, such as a particle filter, due to itshigher robustness [4] and efficient indexing of DEM val-ues. The PMF filter equations are given by:
p(xt) =∑
x
p(xt|xt−1, ut)p(xt−1) (14)
p(xt) = αp(yt|xt)p(xt) (15)
where p(xt|xt−1, ut) describes the vehicle motion prob-ability distribution, p(xt−1) is the prior belief of the ve-hicle state, p(yt|xt) is the likelihood function given inEquation 12, and α is a normalization constant.
3 Baseline TRN System with AUV-Based Maps
The map resolution studies performed in this paper arebased on data collected from two sets of field trials per-formed in 2008 on the MBARI Dorado mapping AUVin Monterey Bay, California. The mapping vehicle, pic-tured in Figure 1, is a high-performance system utilizedby MBARI primarily for generating bathymetry mapswith up to 30cm vertical precision and less than one me-ter lateral resolution [16]. It is equipped with severalsonar systems, including a 200kHz multibeam sonar,two sidescan sonars and a subbottom profiler. The navi-gation system consists of an integrated Kearfott INS andRDI 300 kHz DVL which provides an inertial drift rateof 0.05% of distance traveled, provided the DVL main-tains bottom lock throughout the mission [17].
The terrain navigation algorithm described in Section2 was implemented using the DVL sonar measurementsfrom the vehicle’s navigation system for terrain correla-tions. For these trials, 1m resolution DEM maps wereused, previously generated by the mapping AUV. Utiliz-ing these high-resolution maps in combination with the
Figure 1: Picture of retrieval of the MBARI Doradomapping AUV. c©MBARI.
mapping AUV’s high-performance navigation allows forthe creation of performance baselines for the describedTRN system, necessary for the performance studies inSection 4. The precise navigation system is particularlyimportant for the presented studies as it allows for sepa-ration of performance effects due to sensor accuracy andthose purely due to terrain map accuracy. For these fieldtrials, both the resulting TRN estimation bias and vari-ance are used as performance benchmarks for the analy-sis in Section 4 over coarser map resolutions.
On both missions, the TRN filter was run in real-timeon-board the vehicle at an update rate of 0.1Hz, resultingin filter updates every 15 meters. This lower rate waschosen to minimize the load on the CPU, a LiPPERTCool SpaceRunner 2 with 300MHz processor speed and128MB SDRAM. Even at this lower rate, however, theterrain navigation algorithm successfully converged forboth field trials.
The first set of trials was performed in July, 2008 atSoquel Canyon in Monterey Bay. Figure 2 shows theDEM map along with both the inertial estimate and ter-rain navigation estimate of the vehicle trajectory. Thetrajectory was chosen along this narrow strip of themapped canyon in order to ensure that the vehicle wouldstart at a shallow enough depth to maintain DVL lock,and thus precise navigation, for the duration of the mis-sion. The 90% confidence ellipsoids of the TRN esti-mate are also shown. Figure 3 shows the filter estimatesfor North, East and Depth versus distance traveled. TheTRN filter converges after approximately 30 measure-ments, or 400 meters.
The second set of trials was performed in August,2008 at Portuguese Ledge in Monterey Bay, the samesite as previous trials described in [8]. The DEM map,inertial estimate and TRN estimate of vehicle trajectoryare shown in Figure 4. This vehicle route was selectedto maximize the variability of the terrain under the vehi-cle path, enabling faster estimator convergence than was
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Soquel Canyon 1m AUV Map with Vehicle Route and TRN Estimates
Figure 2: Soquel Canyon high resolution (1m) map withboth Inertial and TRN estimated vehicle tracks. 90%confidence ellipses are shown around the TRN estimate.Figure generated using MB-System c©.
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Figure 3: TRN estimator results in North, East andDepth for the Soquel Canyon field trial data.
achieved in the prior trials at this site. The vehicle wasable to acquire DVL lock for the entire mission due tothe shallowness of this site, ensuring high-performanceinertial navigation. The terrain navigation estimates ver-sus distance traveled are shown in Figure 5. The estima-tor converges rapidly upon reaching the rougher part ofthe terrain, after just 20 measurements or 300 meters.
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Portuguese Ledge 1m AUV Map with Vehicle Route and TRN Estimates
Figure 4: Portuguese Ledge high resolution (1m) mapwith both Inertial and TRN estimated vehicle tracks.90% confidence ellipses are shown around the TRN es-timate. Figure generated using MB-System c©.
Table 1 shows the final TRN estimate variance andbias from inertial for both field trials.
Table 1: TRN Estimates for Field Trials using 1m Reso-lution Maps
North(m) East(m) Depth(m)Soquel Bias -2.94 21.6 -0.65Canyon σ 2.48 1.9 0.16
Portuguese Bias 11.15 -12.54 0.51Ledge σ 3.22 2.74 0.05
Since the terrain navigation filter provides a terrain-relative position estimate, the resulting bias is a com-bination of both inertial navigation error and map geo-registration error. The inertial drift component of thenavigation error is expected to be small for these tri-als due to the high-precision INS unit, at most 1-2mfor both trials. Surface initialization errors in the iner-tial navigation may also exist, however, which may alsocontribute to the observed bias. The expected magni-tude of the geo-registration error was assessed by us-ing a 2D correlation technique to register the coarse
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m)
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90% confidence interval
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TRN MMSE Bias
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100
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t (m
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Dep
th (
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Figure 5: TRN estimator results in North, East andDepth for the Portuguese Ledge field trial data.
ship-based maps from Section 5 with the high-resolutionAUV-based maps for both sites. The resulting expectedgeo-referencing errors are shown in Table 2.
Table 2: Estimated Geo-Referencing Errors for AUV-Based DEMs
North(m) East(m)Soquel Canyon 1.1 11.3
Portuguese Ledge 17.4 -13.3
Given this combination of expected error sources, theresulting magnitudes of the TRN estimation biases in Ta-ble 1 are very reasonable.
4 Map Resolution PerformanceStudies
Utilizing coarser terrain maps for terrain relative naviga-tion is expected to degrade the overall achievable accu-racy, with the degree of performance reduction depend-ing on several factors including both terrain variabilityand vehicle trajectory. In order to evaluate this perfor-mance trade-off, both analytical and empirical perfor-mance studies were completed using the recorded datafrom the field trials discussed in the previous sectionover a set of generated coarse terrain maps.
The coarse maps used in these studies were generatedusing the same processed multibeam sonar soundingsused to generate the 1m resolution AUV maps utilized
in the online field trials. The open-source software MB-System c©was used to generate lower resolution mapsfrom this sonar data. The employed gridding functioncomputes each DEM grid cell value by a Gaussian-weighted average of the sonar soundings falling withinthat cell. This map generation method was used be-cause it is the same method employed to generate theship-based bathymetry maps, thereby resulting in morecomparable DEMs than simple sub-sampling or regularaveraging. For the following studies, the coarse mapswere generated for both Portuguese Ledge and SoquelCanyon at resolutions of 2m, 5m, 10m, 15m, 20m, 25mand 30m.
4.1 Empirical Performance Studies
In order to assess TRN performance with decreased mapresolution, experimental tests were performed first byrunning the terrain navigation filter over the generatedlow-resolution DEMs for both field data sets. Figure6 shows the resulting filter performance for the SoquelCanyon data, shown as root mean squared error (RMSE)for both North and East. The RMSE is further decom-posed into bias and variance. The bias is computed asthe deviation in the TRN estimate from the nominal es-timates over 1m resolution maps, shown in Table 1. Fig-ure 7 shows the equivalent results for the PortugueseLedge data set.
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8TRN performance over varying map resolution − Soquel Canyon
Nor
th (
m)
RMSE
Bias
Std Dev
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4
6
Map Resolution (m)
Eas
t (m
)
Figure 6: Root mean square error of the final TRN esti-mate for varying map resolution at Soquel Canyon. Re-sults are shown as total RMSE as well as the componentbias and standard deviation for both North and East.
The results from these experiments show that the in-crease in RMSE of the TRN estimator with decreasedmap resolution occurs gradually. Furthermore, despitedifferences in the relative decomposition of bias and
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6TRN performance over varying map resolution − Portuguese Ledge
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Bias
Std Dev
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t (m
)
Figure 7: Root mean square error of the final TRN es-timate for varying map resolution at Portuguese Ledge.Results are shown as total RMSE as well as the com-ponent bias and standard deviation for both North andEast.
variance for the two data sets, the overall rate of per-formance degradation is highly comparable. In order tounderstand the underlying cause of these observed simi-larities and differences between the two terrains, analyt-ical studies were performed for both data sets, which arediscussed in the following section.
4.2 Analytical Performance StudiesA posterior Cramer-Rao Lower Bound (PCRLB) analy-sis was utilized to provide an analytical bound on TRNperformance over the generated low-resolution maps.The PCRLB is an extension of the common Cramer-RaoLower Bound (CRLB), which provides a lower boundon the mean square error of any unbiased estimator fortime-invariant systems. The CRLB is defined as theinverse of the Fisher Information matrix, J , computeddirectly from the measurement probability distribution(e.g. [18]):
CRLB = J−1 ≤ E[[x− x][x− x]T
](16)
where
J = −E[∂2 ln p(y|x)∂xi∂xj
]. (17)
For dynamic systems where the state exhibits uncertainchange, a posterior Cramer-Rao bound can be found byutilizing the full Fisher information matrix over the jointprobability distribution of the measurement and state:
Jn = −E[∂2 ln p(yn, xn)
∂xi∂xj
]. (18)
While applicable to many systems, the PCRLB is par-ticularly powerful as a performance assessment tool fornon-linear filters, such as the point mass filter used inthis paper. Tichavksy et al. [19] developed a very usefulformulation of this bound specifically for discrete-timenonlinear filters. The authors show that Jn can be cal-culated via the Ricatti-like recursion shown in Equation19
Jn+1 = D22n −D21
n (Jn +D11n )−1D12
n (19)
where
D11n = E
[− ∂2
∂x2n
log p(xn+1|xn)]
D12n = E
[− ∂2
∂xn∂xn+1log p(xn+1|xn)
]D21
n = [D12n ]T
D22n = E
[− ∂2
∂x2n+1
log p(xn+1|xn)]
+E[− ∂2
∂x2n+1
log p(zn+1|xn+1)]
= D22n1 +D22
n2. (20)
For systems with additive Gaussian process and mea-surement noise, as with the TRN system, Equation 19simplifies significantly with D11
n = D12n = D22
n1 = Σ−1r,n
and D22n2 = J , the Fisher Information matrix for the
standard CRLB. In this simplified case, the PCRLB isthe covariance of the Extended Kalman Filter.
Both versions of the Cramer-Rao bound have beensuccessfully applied to performance assessment in ter-rain navigation systems, with the CRLB being used forbatch TRN systems [3] and the PCRLB for non-linearBayesian TRN implementations [5], [20]. The PRCLBfor the current TRN filter implementation described inSection 2 is identical to that derived by Bergman [20],with a modified J term for the likelihood function de-scribed in Equation 12. Using the Matrix Inversionlemma to rearrange Equation 19, the resulting PCRLBfor the employed TRN filter is given by:
PCRLBn = Jn−1 ≤ E
[[xn − xn][xn − xn]T
](21)
where
Jn+1=HTn+1ΠT
n+1Σ−1w,nΠn+1Hn+1+(J−1
n +Σr,n)−1 (22)
and Hn = ∇xh(xn) is the terrain gradient matrix forthe current measurement. Note that the terrain gradientsin Hn are the gradients seen by the estimation filter, notthe true gradients of the physical terrain. Thus this ma-trix is determined by both the utilized DEM map and theemployed interpolation method.
Figure 8 shows the result of applying the PCRLB inEquation 22 to the Soquel Canyon data set over the gen-erated coarser maps. The root mean square error fromthe empirical studies is shown as well for comparison.The corresponding results are shown in Figure 9 for thePortuguese Ledge field data.
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Terrain Navigation Root Mean Square Error (RMSE)with Varying Map Resolution − PCRLB vs. Actual
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MS
E (
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Actual
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SE
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)
Figure 8: Posterior Cramer-Rao Lower Bound for theSoquel Canyon data set shown along with the experi-mentally determined RMSE.
For both data sets, the estimator performance trendfollows that of the PCRLB. For Soquel Canyon, the gapbetween estimator RMSE and the PCRLB is noticeablylarger than that for Portuguese Ledge, a result of thelarger biases observed in the estimator for coarser mapresolutions.
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Terrain Navigation Root Mean Square Error (RMSE)with Varying Map Resolution − PCRLB vs. Actual
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MS
E (
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5
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SE
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Figure 9: Posterior Cramer-Rao Lower Bound for thePortuguese Ledge data set shown along with the experi-mentally determined RMSE.
While the PCRLB is not an exact predictor of esti-
mator performance (i.e., the performance bound is notmet for this data), it can still provide a meaningful toolfor assessing the sources of performance variability forthese different terrains. From Equation 22, reduced TRNestimator performance can be attributed to two primaryfactors: (1) decreases in H , corresponding to reducedvariability in the represented terrain gradients, and (2)increases in Σw, corresponding to decreased accuracyof individual DEM points. Thus to better understand theperformance similarities and differences noted in Sec-tion 4.1, comparisons of these two factors were madefor the two sets of field trials.
Figure 10 shows a comparison of terrain gradientstatistics for the Portuguese Ledge and Soquel Canyonmaps for varying resolutions. The terrain slopes used forthese statistics were computed for the section of terrainalong the vehicle path. For both terrains, the slope mag-nitude and variability is seen to decrease primarily from1m to 5m, and then level off for coarser map resolutions.This leveling behavior mimics the concave, gradual in-cline of the PCRLB curves seen in Figures 8 and 9.
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Mea
n (o )
Terrain slope statistics along vehicle path
Portuguese Ledge
Soquel Canyon
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Figure 10: Comparison of terrain slope statistics alongvehicle path for Soquel Canyon and Portuguese Ledge.
Figure 10 further illustrates that the Soquel Canyonterrain maps exhibit both significantly larger overallslope variability and slope magnitude than the Por-tuguese Ledge maps for nearly all map resolutions. Thischaracteristic alone suggests higher performance TRNfor the Soquel Canyon terrain. However, as noted earlier,the performance is also effected by the DEM accuracy.
DEM accuracy is represented in Σw by the σ2v term,
defined in Equation 8. The magnitude of σ2v depends on
both the accuracy of the DEM generation process andthe underlying variability in the terrain, represented bythe variogram, γ(s). Indeed, the RMSE of interpolatedDEM values has been shown to be a direct function ofthe variogram [11]. Thus for the same quality under-lying DEM measurements, the accuracy of interpolated
DEM elevations will be larger for terrains with largervariograms.
Figure 11 shows the variograms for the SoquelCanyon and Portuguese Ledge terrains over the rangeof map resolutions considered in these studies. Thevariograms were computed using the fractal model de-scribed in Equation 8.
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8
10
||h|| − Map point spatial separation (m)
Var
iogr
am γ(
||h||)
Variogram comparison
Portuguese Ledge
Soquel Canyon
Figure 11: Comparison of Variograms
Note that the variogram for Soquel Canyon increasessignificantly with lag distance, compared to a muchmore gradual incline for Portuguese Ledge. As a result,the Soquel Canyon data is expected to exhibit notice-ably reduced DEM accuracy at coarser resolutions. Thisreduced map accuracy combined with large slope vari-ability noted in Figure 10 makes the Soquel Canyon ter-rain much more susceptible to terrain model errors thanthe Portuguese Ledge data set, resulting in the larger ob-served estimator biases. In addition, however, the largeterrain gradients observed for the Soquel Canyon databalance the reduced DEM accuracy in terms of the esti-mator variance, predicted by the PCRLB.
TRN performance reduction with coarser maps canthus be ascribed to the decreased ability of the resultingterrain model to accurately represent both the true slopevariability and elevation. For flatter terrains, the per-formance reduction will primarily result from reducedDEM slope variability. Conversely, TRN over terrainswith larger slope and elevation variability will be ef-fected primarily by reduced DEM elevation accuracy.Thus in many physical terrains, the effect of H and Σw
in the PCLRB are expected to counter-balance, resultingin comparable TRN performance reductions regardlessof the underlying terrain.
5 TRN with Ship-Based Maps
While the studies in the previous section are insightful,it is important to assess how well this analysis on artifi-cially coarse maps matches with truly coarse DEM maps
generated from ship-based bathymetry. As noted ear-lier, ship-based bathymetry is available for a large sec-tion of Monterey Bay, including the sites for both of thepresented field trials. Using this ship-based multibeamsonar data, 20m resolution DEM maps were generatedfor both Portuguese Ledge and Soquel Canyon usingthe same MB-System c©gridding method as described inSection 4.
Figures 12 and 13 show the DEM maps for SoquelCanyon and Portuguese Ledge, respectively, overlaidwith the terrain navigation estimated vehicle tracks and90% confidence ellipses. The corresponding final esti-mation bias and variance are shown in Table 3, alongwith the PCRLB performance bound computed in Sec-tion 4.2 for a 20m resolution map.
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Soquel Canyon 20m Ship−Based Map with Vehicle Route and TRN Estimates
Figure 12: TRN estimator results for Soquel Canyonfield trials, using ship-based, 20m resolution bathymetrymap. Figure generated using MB-System c©.
Note that since these maps are generated from shipswhich can maintain gps lock during data acquisition, thecontribution of the geo-referencing error to the resultingTRN bias is expected to be small. However, initializa-tion errors in the inertial navigation are still plausible.In addition, TRN estimation bias on the order of a cou-ple meters is expected simply due to the decreased mapresolution, observed in Section 4.1.
For both of these data sets, the observed estimatorvariances are very comparable to the predicted estima-
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Portuguese Ledge 20m Ship−Based Map with Vehicle Route and TRN Estimates
Figure 13: TRN estimator results for Portuguese Ledgefield trials, using ship-based, 20m resolution bathymetrymap. Figure generated using MB-System c©.
Table 3: TRN Estimates for Field Trials using 20m Res-olution Ship-Based Maps
North(m) East(m) Depth(m)Soquel Bias -7.31 14.86 0.31Canyon σ 4.53 3.64 0.23
RMSE 8.6 15.3 0.38√PCRLB 3.79 3.03 –
Portuguese Bias -3.15 0.03 0.3Ledge σ 5.16 3.91 0.03
RMSE 6.05 3.91 0.3√PCRLB 4.09 3.75 –
tion performance provided by the PCRLB. In addition,the resulting estimation bias from the inertial naviga-tion is reasonable given the expected bias sources notedabove. This similarity of predicted and actual perfor-mance indicates that the presented map resolution stud-ies in Section 4 are realistic predictions of TRN perfor-mance on actual coarse DEMs.
6 Conclusions
The presented performance studies in this paper indi-cate that, while TRN performance does degrade withdecreased map resolution as expected, this reduction oc-curs gradually. Moving from a 1m to a 30m resolutionmap was observed to decrease estimator accuracy from∼ 2m to ∼ 6m RMSE. A posterior Cramer-Rao bound
analysis demonstrated that the observed degree of per-formance reduction is primarily related to two counter-acting characteristics in the underlying terrain: variabil-ity in terrain slope and elevation.
For a fairly flat terrain, reduction in map resolutionsignificantly decreased the overall terrain gradient dis-tribution, resulting in reduced achievable TRN perfor-mance. However, this reduction was counteracted bythe terrain’s small elevation variability, resulting in min-imal DEM inaccuracies at coarser resolutions. In a moresteeply sloped terrain, these effects were found to beopposite. It is expected that these two characteristicswill naturally compete in any terrain surface, resulting incomparable performance reduction rates for many natu-ral terrain surfaces.
The results in this paper thus suggest that readilyavailable ship-based bathymetry can provide sufficientterrain information to achieve TRN accuracies less thanthe map resolution. While the use of high-resolutionAUV-based maps will provide improved performancewhen available, such maps are not crucial to successfulTRN and may be unnecessary for many applications.
AcknowledgmentsThe authors would like to thank MBARI for support andthe Stanford Graduate Fellowship for partial funding ofthis work. Researchers at MBARI who were particularlyhelpful in acquiring the presented data include Dave Ca-ress, Hans Thomas, and Brian Schlining.
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