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HausdorffBased RC and IESIL Combined Positioning Algorithm for Underwater Geomagnetic Navigation
EURASIP Journal on Advances in Signal Processing volumeÂ 2010, ArticleÂ number:Â 593238 (2010)
Abstract
This paper presents a primitive solution with novel scheme and algorithm for Underwater geoMagnetic Navigation (UMN), which now occurs as the hotpoint in the research field of navigation. UMN as an independent or supplementary technique can theoretically supply accurate locations for marine vehicles, but in practice there are plenty of restrictions for UMN's application (e.g., geomagnetic daily variation). After analysis of the theoretical model of geomagnetic positioning in the correlationmatching mode from the viewpoint of pattern recognition, this paper proposed an appropriate matching scenario and a combined positioning algorithm for UMN. The subalgorithm of Hausdorffbased Relative Correlation (RC) corresponding to the pattern classification module implements the coarse positioning, and the subalgorithm of Isograms EquidistanceSegmenting theIntersection Lines (IESILs) associated with the module of feature extraction continues the fine positioning. The experiments based on the simulation platform and the realsurveyed data both validate the new algorithm, and its efficiency and accuracy are also discussed. It can be concluded that the work introduced in this paper gives an initial and real validation of UMN's potentiality.
1. Introduction
The technique of Underwater geoMagnetic Navigation (UMN) [1] recently has become a hotpoint in the research area of navigation. This trend will develop further with the unreliability of the Global Navigation Satellite System (GNSS) increasing [2], and with the requirements growing in some special conditions (e.g., submarine exploration) [3]. It even has been deduced that only the Geophysical Field of Earth (GFE) referencing methods (UMN is a typical one) can enable Unmanned Underwater Vehicles (UUV) to navigate accurately over large areas [4, 5]. The sea turtles using geomagnetic fields for navigation [6â€“9] seem to convince people more to apply UMN. Moreover, as an important supplementary means, UMN can also supply an aided solution for rectifying Inertial Navigation System's (INS's) errors which accumulate with time [10]. UMN also can be integrated with other methods [11] to improve the accuracy of marine navigation. Overall, all influences factors causing voyage drift (such as water currents or sonar failure) are taken for granted in a "blackbox" in this paper, and UMN as an independent methodology will be explored to supply the accurate positioning information to rectify the whole "blackbox."
After analysis of the positioning algorithms adapted or established for UMN after the literature review, it can be summarized that batchcorrelation is one kind of fundamental schemes [12â€“14]. The basic idea is to match the realtime sampling sequence with the pickedout geomagnetic characteristic sequences from the presurveyed maps (called formatching sequences). The formatching sequences are often of total intensity, and one of them is corresponding to the reallocation sequence. The current locations of vehicles can be determined inversely by seeking the extreme of the correlation function, while the estimated "hunting window" (e.g., according to INS's parameters) can reduce the traversal range for extracting the formatching sequences [15].
But the traditional positioning algorithms developed for UMN cannot let Marine geoMagnetic Field (MMF) play its full role as a good locationreferencing source. Currently the oftenâ€”referred correlationâ€”type algorithms include the Vector Searching algorithm (VS) [12] adapted from the TERrain COntour Matching (TERCOM) algorithm originating from terrainaided navigation [16], the improved ICP (iICP) algorithm [13] transformed from the Iterative Closest Point (ICP) algorithm initially for images registration [17], and so on. But each of these algorithms applied on MMF still suffers from some limitations. In the critical conditions which allow only onetime matching, VS even cannot supply the relatively accurate headingangle. iICP is degraded by its complex matrix transformation of shift and rotation. Therefore, a more effective positioning algorithm should be built with consideration of MMF's inherent properties.
Simultaneously there are also plenty of practical restrictions hampering UMN's application (e.g., the disturbances caused by geomagnetic storm). The inherent magnetic fields of marine vehicles sometimes can locally disturb MMF, and this may make magnetometers sample the false information which is supposed to reflect MMF's real distribution. Additionally, geomagnetic fields have daily, monthly, and even annual variations. These are another kind of large influence factors to UMN and sometimes even make UMN unavailable. The improvements on hardware cannot solve all these issues. So, the positioning algorithms need some extra procedures to tolerate these undesirable factors, or even some novel and robust algorithms shall be developed from the beginning.
Consequently, from the viewpoint of pattern recognition, the theoretical models of the positioning algorithms in the matching mode are analyzed in an innovative way. Based on these analyses, this paper proposed an appropriate matching scheme and a combination positioning algorithm for UMN. The subalgorithm of Hausdorffbased Relative Correlation (RC) as the pattern classification module implements the coarse positioning and can overcome the influences of geomagnetic variations somehow. The subalgorithm of Isograms EquidistanceSegmenting the Intersection Lines (IESILs) related to the feature extraction module continues the fine positioning, which is based on the accustomed cruisingstate. The state means that marine vehicles pass through the designated MMF reference areas in the Uniform Linear Motion mode (ULM) intentionally, to acquire more reliable locationassociated geomagnetic information.
This paper is organized as follows. After Section 1 introducing the UMN research background, the second section makes premises analysis for a highperformance algorithm, by reviewing the traditional algorithms, generalizing UMN's restrictions, and analogizing the scenario of pattern recognition. Section 3 presents the new positioning algorithm comprising Hausdorffbased RC and IESIL. After the experimental validations in Section 4, Section 5 will go on with some discussions about the algorithm's accuracy and efficiency. Then some conclusions are achieved.
2. Premises Analysis
To establish a positioning algorithm appropriate for UMN with high performance, it is necessary to clarify the shortages of the traditional algorithms and to grasp the specialties of MMF. After these premises are made clear, they can work as the cutpoints for a new and suitable positioning algorithm. Besides, a more proper scheme based on MMF's properties can be instructive to a more powerful algorithm. The following work in fact is an expansion of threeaspect review of traditional algorithms, analysis of MMF's specialties, and proposal for a new scheme.
2.1. Algorithms Review
From the Introduction, it can be learnt that the two typical positioning algorithms for UMN in batchcorrelation mode are VS and iICP. Therefore, the reviews on these two algorithms can representatively give the shortages of the algorithms of this kind.
The brief idea of VS [12, 14] is based on the assumption of the formatching sequences parallel with the continuously recorded trajectory (usually assuming the one indicated by INS). This simplification will introduce some errors into the matching results inherently, because INS's output of heading angle generally deviates from the real one. Continuous matching may make over this fault to some extent, but in the special conditions which urgently allow for only onetime matching or suffer from large measurement errors, the results after this method even cannot satisfy the requirement of mediumlevel accuracy during navigation.
The scenario of iICP [13] is by transforming recursively the temporary trajectory, which is connected by the projected points on the corresponding isograms, to approximate the real one. Its brief procedures contain large matrix transformation, singular value decomposition, orthogonal matrix calculation, and so on. These complex operations as well as the distortion of isograms sometimes lead to the embarrassment of solution loss. iICP is complicated and is not desirable for UMN under the demanding of immediate positioning, and the rasterized storage of MMF maps with large grid spacing may add the difficulties more.
From the perspective of algorithm implementation, the new algorithm for UMN needs to avoid the ill effects of parallel assumption in VS and complex transformation in iICP. The omission of the yawing errors in VS can be overcome by choosing the formatching sequences in all possible directions, and the excess of the projecting operations in iICP shall be reduced by restricting each formatching sequence into a line. The first and last isograms in the "hunting window" can reduce the possible directions into a reasonable extent, and marine vehicles generally can keep cruising in a line. Hereupon, modifying the positioning algorithms with some unused conditions such as cruise pattern is possible to improve their performance.
2.2. Restrictions Analysis
The factors influencing UMN remarkably include marine vehicles' own magnetic fields, the characteristic diversity of MMF maps, and the fluctuations of geomagnetic fields. The first two issues can be solved by the draggingmeasurement mode and optimalpath planning separately. But for the third issue, there has been no ideal method to overcome it before.
Geomagnetic field itself has the shorttime disturbances (e.g., daily variation) and the longterm fluctuations (e.g., monthly variation and even annual variation). The daily variation is illustrated in Figure 1, and longterm variation is exemplified in Figure 2 (a typical area with a larger magnitude of fluctuations). The daily variation makes the values of the realtime sampling sequence deviate from the presurveyed geomagnetic maps (mostly after the process of variation normalization already), with the maximum difference value up to 40 nanoTelsa (nT) indicated in Figure 1. The longterm variation makes the realtime sampled intensity away from its presurveyed maps also, with an accumulated deviationvalue about 20â€‰nT for half a year as showed in Figure 2. Now neither can the daily variation be modeled accurately, nor can the geomagnetic variations be monitored effectively during voyage. All in all, this kind of disturbing factors can affect the positioning results with large errors and sometimes exerts the traditional correlationbased positioning algorithms fail.
In terms of the premises that samplings are limited to a certain number for each time matching, the aforementioned deviations theoretically can be simplified into a constant value. As alluded in Figure 1, the underlying basis of this simplification can be basically proved by randomly extracting time intervals with several tens of minutes, in which daily variation almost keeps constant with a very little change. Namely, the measurement sequence has a constant but valueunknown deviation compared to the formatching sequences (also with other lowlevel noises). The simplification is also available for monthly and annual variations. Then the settlement is to search an effective matching criterion to judge the coincidence of the measurement sequence and the formatching sequences, between which there is a constant difference in priority but the value of the constant is unknown. Actually, this constantdominant difference restricts the algorithms of recursive filtering kind [19, 20], which have been also tried on MMFs, mostly still in the phase of methods adaptation under relatively ideal experimental environments with little noises now.
2.3. Scheme of Pattern Recognition
Sections 2.1 and 2.2 actually summarize the practical problems into two premises for an effective positioning algorithm: Premise 1 is how to assure the realcoursecorresponded characteristic sequence lying in the formatching sequences; Premise 2 is about how to choose the criterion function overcoming the influences of geomagnetic fluctuations. Consequently these require us to consider the matching process more synthetically and to seek a new scheme as the theoretical basis for a more adaptive and comprehensive algorithm.
As we know, the scheme of pattern recognition presents an efficient frame for many judgmentoriented problems. In fact, the positioning process by batchcorrelation can also be classified into the research area of pattern recognition, as in how to approach the true objects (location here). Pattern recognition generally contains twomodule feature extraction and pattern classification, and positioning can also be divided into these two stages. Feature extraction corresponds to the process of searching all the formatching sequences, and pattern classification is related with the step of choosing the optimal criterion function to determine the real location.
The theoretical framework, the functional operations, and the concrete algorithms are listed in Figure 3, which gives an explicit scenario of the matching process for UMN positioning algorithms. The mathematical expressions of these two steps are generalized into
where means the objective characteristic sequence, which corresponds to the most accurate positioning results. Function (1) denotes the necessary condition for the optimal positioning algorithm, and the real trajectory must be, or close to, one of the formatching sequences by . Equation (2) solves it with the goal function .
After analysis based on this frame, it can be studied where the restrictions of the traditional matching algorithms lie. VS focuses on the module of pattern classification, while iICP pays more attention on the function of feature extraction. The scheme in pattern recognition gives a more effective plan, which is to improve the two modules profoundly at the same time.
Accordingly, with the scheme for matching from the viewpoint of pattern recognition displayed in Figure 3, it can be clarified that Premise 1 is involved with the module of feature extraction and Premise 2 is associated with the module of pattern classification. They are corresponding to Sections 2.1 and 2.2, respectively, and the following new algorithm is proposed by overcoming the problems mentioned in Sections 2.1 and 2.2 separately.
3. Combination Algorithm
The novel combination algorithm comprises two kernel subalgorithms: The subalgorithm of Hausdorffbased RC corresponds to the pattern classification module, and the subalgorithm of IESIL works as the feature extraction module. IESIL is introduced firstly, as it can somehow avoid the limitations embedded in the traditional correlationtype algorithms.
3.1. IESIL
Given the course between the two endpoints of the measured sequence for positioning (named EEcourse), IESIL theoretically is to approximate the ideal operation of equidistancesegmenting the EEcourse. This is realized by the isograms corresponding to the realtime samplings, which are collected with equaltime intervals. Therewith, the basic scheme is as follows. Firstly, in the "hunting window" estimated with INS's parameters, the measurementscorresponded isograms are extracted; secondly, searching the intersection line with Minimum Standard Deviation (MSD) about the lengths of its line segments, which are divided out by the isograms. And then the accurate location and attitude information can be obtained inversely. All the processes are considered in the regular grid form for convenience of digital processing, and this also fits the storage mode of most MMF source data. The concrete steps are as follows.
Step 1 (Isograms extraction).
As MMF maps are generally stored in grid form, tracing isograms can basically be considered just within each grid cell. Suppos that the four points comprise one little cell, in which serves as the bottomleft vertex and the other three distribute clockwise sequentially with as their coordinates and as the characteristic values correspondingly. Then the value of any interior point can be gained by the interpolation methods, such as bilinear interpolation with
where and . Inversely, the contour relating to a given characteristic value in cells can be expressed by
Equation (4) indicates that the isograms in the cells are of hyperbolic form, which will consume a lot of computation resources. So the point set satisfying on the four sides of each cell is alternatively searched, and then the sought points (called Ipoints) can be connected to construct the isograms, respectively. This simplification may introduce some little error into the positioning results but can significantly reduce computation time.
Step 2 (Formatching sequences determination).
The aforementioned formatching sequences actually need to satisfy two conditions: (a) the whole length between the first and last isograms is equivalent to the related EEcourse ; (b) the sequences must cross all the isograms got in Step 1. If these two requirements are satisfied, the associated intersectionline can be integrated into the formatching sequences. The detailed judgment includes two parts as follows.
Firstly, the first and last crossingpoints of each intersectionline are calculated. The first point assumes one Ipoint of the first contour determined by Step 1. Then the circle with the first point as center and as radius is constructed, and the last point can be sought as the crossing point (possibly one with several ) between the circle and the last isogram. The process can be expressed by (5). The first and last crossing points (named EEpoints) are thus determined and added into the chain list of EEpoints. Then another Ipoint of the first isogram is picked and the same process is iterated
Secondly, based on each pair of EEpoints obtained above, the function of related intersection line can be determined, and the crossing points with the remaining isograms in middle (called Mpoints) can be acquired with (6). If all the remaining isograms are crossed, the EEpoints and associated Mpoints will be joint into one formatching sequence. If one contour has no crossing point in the hunting window, the calculation of Mpoints can be stopped for the following contours, and the crossing points calculated before on this possible intersection line must be abandoned. The same operations are enforced iteratively for the whole chain list of EEpoints, and the set of formatching sequences will be generated
Step 3 (Minimum MSD calculation).
Seeking the minimum MSD about the lengths of the line segments, which is based on (7), is the concrete mathematical implementation of the isograms equidistancesegmenting the intersection lines. The location relating to the minimum MSD will be the most possible positioning result. The heading angle can also be solved by , and will be valued with under the special condition
Step 4 (Recompare iteratively with halfspacing).
After the intersection line with the minimum MSD is achieved, the two points and with distance of half grid interval to the first point on the same contour are picked out. Then Steps 2 and 3 are iterated on these two points. (A) If the minimum MSD related with points and is larger than , the point can be determined as the real position. (B) Otherwise, or with less MSD replaces , and the STEPs are iterated from or until the condition (A) is satisfied. The heading angle can be resolved. In case of excessive iteration, the number of times for approximating the new first point by and is restricted with a given threshold value.
The four steps depicted above can ultimately output the solved location and heading angle of marine vehicles.
3.2. HausdorffBased RC
IESIL can somehow diminish the issues listed in Section 2.1. But for the restrictions in Section 2.2, IESIL cannot overcome those problems just by improving the feature extraction module. The deviation of the positioning results caused by geomagnetic fluctuations shall be reduced in the module of pattern classification by exploring MMF's properties.
This subsection explains RC with the traditional criteria, such as Mean Absolute Difference (MAD) and Mean Square Difference (MSD) in (8). It is theoretically proper for MMFs. The idea of RC is to subtract the corresponding mean values from all formatching sequences and then calculate the correlation degree between them and the measurement sequence also with its average removed. With this preprocess, the constantlike deviation of realtime samplings caused by daily variation or other variations can be omitted during the matching. As matching for each time takes just a certain number of samplings, both averages removal can overcome the influences of daily variation and so on
where and are the measurement sequence and formatching sequence with their averages and removed. The minimum correlation results give the real trajectory, and the new judging function with the improved criterion of Hausdorff distance is formulated into
where with the extremesearching subfunctions of and .
Hausdorff distance reveals the mismatch degree between the characteristic sets and . Namely, a larger Hausdorff distance denotes the greater difference between two characteristic sequences. Its advantage is insensitive to lowlevel noises. Thus, with the improvements of Hausdorff distance to overcome the measurement random noises and RC to reduce the whole deviation produced by geomagnetic fluctuations, the subalgorithm of Hausdorffbased RC supplies a good settlement to MMF's own disturbances.
3.3. Procedures
Based on the two subalgorithms established above, this subsection presents a full scenario of the proposed positioning algorithm. Hausdorffbased RC is carried out for coarse positioning, and then IESIL is deployed for further fine positioning. The procedures of the combination algorithm in Figure 4 are depicted in detail as follows.
Procedure 1.
The set of formatching sequences parallel to the trajectory indicated by INS are extracted, and Hausdorffbased RC is run based on this sequence set for coarse positioning.
Procedure 2.
With the results of coarse positioning, the constantinpriority deviations caused by the influences (e.g., daily variation) can be rectified from the measurement sequences.
Procedure 3.
Based on the results of the coarse positioning and the measurement sequences after rectification, IESIL completes the fine positioning. The process takes the coarse location as the center of the hunting window, and the real location can be achieved iteratively.
4. Experiments
The experiments are firstly conducted on our simulation platform to testify the new algorithm's applicability to the diversity of the geomagnetic reference maps, and some experiments are also tried on the realsurveyed MMF data to validate the algorithm in real environments.
4.1. Simulation Experiments
The simulation experiments were executed on some local geomagnetic maps. The positioning results in Figure 5 give an illustration of the positioning accuracy of the combination algorithm. The related MMF map is stored in grid form with points and is collected in geomagnetic total intensity. The compositional anomaly intensity is between 1.4 and 265.2â€‰nT. The display module assumes the grid spacing as the coordinate unit (named gridinterval).
The experimental conditions were configured as follows. The initial location indicated by INS is (12, 12), and the yawing angle is to north. The distance between two samplings is 5.7 gridintervals, and the sampling number accumulates to 10 then to comprise a formatching sequence. The real start point lies in the circle with radius of 2 gridintervals and with center at the start location indicated by INS. The real heading angle is to north, and the sampling interval is equal to INS's one. The errors of samplings are yielded randomly as Gaussian white noises with the standard deviation of 5â€‰nT. The daily variation is simulated with a deviation of 20â€‰nT, which is added to the measurement sequence.
The result of coarse positioning with Hausdorffbased RC is showed in Figure 5(a). It indicates that the resulted trajectory is still parallel with the output one by INS. And the yawing error remains nonignorable. Next, the measurement sequence is modified by setting its initial value equal with the first one of the resulted sequence after coarse positioning. Then IESIL is executed and the result of fine positioning is demonstrated in Figure 5(b), and it approximates the real trajectory with deviation less than 1 grid interval. The results of Hausdorffbased RC and IESIL both lie in the "hunting window". From Figure 5(a) to Figure 5(b), the validity of the combination algorithm can be explicitly judged by the spatial relationships between the coarsely resulted track, fineresulted track, INS track, and real track. IESILresulted fine track is more close to the real track than to RCresulted coarse track, and both are better than the INS track. The rectifying effect for INS's drift is very obvious. The comparison basically confirms the anticipation that the combination algorithm can supply a more effective method for UMN.
4.2. Real Data Validation
The disturbances in practical environments are more complex than the assumptions of noises in the simulation platform. To validate the combination algorithm, the postprocessing experiments based on the real MMF data are executed. The surveying of the MMF data was accomplished with recording of INS and GPS data synchronously at Bo Sea from June 23, 2008 to August 17, 2008.
4.2.1. RealSurveyed Data
The surveying conditions at Bo Sea were planed as such the region occupies 10 kilometers in the westeast direction and 5 kilometers in the northsouth direction. The range of geomagnetic total intensity is (53476, 53642) nT. The location on GoogleEarth is indicated in Figure 6.
The magnetometer assumed is Ocean G882 of cesium opticalpump type. Its sampling frequency is set with 5â€‰Hz, and its measurement resolution is adapted to 0.001â€‰nT. INS uses the Laser INS with the moderate drift of /h. GPS applies the Trimbletype Difference GPS (DGPS) with precision of 0.1â€‰m. The ship's velocity is 5â€“8 knots (1 knot = 1.852 kilometers/hour). The distance between the measurement lines is 200â€‰m, and the sampling points on each line have the average interval of 0.1â€‰m. The boat is mainly of wood as displayed in Figure 7, but it still has about 4 tons of iron materials. So the draggingmeasurement mode with cable length of 50â€‰m is utilized, and the magnetometer floats between the seasurface and 2â€‰m below. If a metaldominant boat uses UMN, maybe a longer cable and a more precise draggingmodel are needed.
The local MMF map constructed from the surveyed data is presented with isograms in Figure 8. certain number of samplings from the diagonal course, which is designed for testing, were picked out as the measurement sequence. The voyage direction is from southeast to northwest. The GPSindicated and INSindicated trajectories are drawn also in Figure 8, and they are chosen on purpose to represent the situation with great INS's drift. Under the relative reference coordinates with (North, East) as the variable, the initial point of GPSindicated trajectory lies in (32.3088, 58.3222), and the initial point of INSindicated trajectory lies in (38.5096, 42.4688). Their last points have locations (43.1052, 37.2662) and (49.3948, 21.6530) individually.
4.2.2. Data Analysis
The statistical analysis of this MMF area manifests that the range of the geomagnetic intensity is (53476, 53642) nT, and its standard deviation is 43.49. The standard deviation suggests that this area statistically has ample geomagnetic fluctuations suitable for UMN, but the distribution trend of geomagnetic intensity is declining southtonorth in a relatively smooth mode. This indicates that the area is not an ideal region with rich spatial referencing characteristics, but for validating the positioning algorithm it can supply a stringent environment.
As showed in Figure 8, the heading bias between INStrajectory and GPStrajectory is small, but the location deviation is large. The distance is 851â€‰m from the last location output by INS to the one indicated by DGPS. This indicates that INS has a great drift and it needs the rectification by UMN. The location sequence from DGPS, which delegates the track of the vehicle, is linear on the whole, and this means that the boat cruises in a line. As a result, the realsurveyed data can be used as the references for validating of the combination algorithm.
The geomagnetic reference map is the result after normalization of daily variation, while the realtime measurement sequence during navigation still keeps this influence factor. The daily variation corresponding to the realmeasurement is segmented out and demonstrated in Figure 9. The normalization value is related with the dotted line. Besides the lowlevel noises during the surveying, there is a nearly constant difference about 15â€‰nT. So, this condition with great deviation is the typical one for experiments, and this presents a more difficult testing to the new algorithm.
To test the combination algorithm's applicability to the diversity of sampling noises, the constant part of daily variation is subtracted from the measurement sequence for comparison. The resulted situation can be used to testify the algorithm in case of nonwhite noises in lowlevel. Upon that, the postprocessing experimental conditions will be separated into two cases for consideration: Status 1, the measurement sequence removed of the large constant deviation serves as the real sequence close to the normalization reference, as the instances at about 0 o'clock nearby in Figure 9; Status 2, the measurement sequence keeping the large constant deviation works as the unstable sequence in reality. The respective distributions are demonstrated in Figure 10, and the following work will be expanded on these two cases.
4.2.3. Positioning Results
In fact the new combination algorithm can overcome the influences of the large deviation, and the results of Status 1 and 2 are equivalent. The positioning results are showed in Figure 11. The deviation between the last locations output initially by INS and DGPS is 851.15â€‰m. The location deviation is rectified to 672.35â€‰m by VS, while it is reduced to 393.63â€‰m after the combination algorithm. The latter result satisfies the requirement posed in the supporting project concerning initial exploration of UMN. Based on the postprocessing on the local MMF data, INS data and GPS data, Hausdorffbased RC and IESIL combined algorithm have been basically validated.
The rectified deviation of 393.63â€‰m is still not ideal and needs further improvements. Combined with Figure 10, it can be learnt that the distribution trend of the left sequence (Status 1) is similar with the real sequence, but obviously there are still some differences. Hence, it is not enough to just relying on algorithms to overcome the incoincidence between the sampled and real sequences, and the real solution for positioning in UMN is still to model the geomagnetic variations accurately.
5. Discussions
5.1. Performance Comparison
The performance comparison is based on the conditions which are identical to the simulation experiments in Section 4.1. Hausdorffbased RC is the expansion of the traditional VS algorithm without much difference in runtime, while IESIL and iICP both based on the isograms generate the results with almost the same accuracy. Thereupon, performance comparison can focus on some special aspects. As the producer of the combination algorithm's last results, IESIL will be assessed by comparing it with VS mainly for the accuracy of positioning and orientating. As the main timeconsumer, IESIL will be compared with iICP briefly for the perspective of efficiency.
Different level of sampling errors commonly corresponds to different positioning accuracy, and so the positioning accuracy of the algorithm is considered along with the fault tolerance. With the noises increasing, the comparison between IESIL and VS is charted in Figures 12(a) and 12(b). Under the lowlevel measurement errors, positioning and orientating by IESIL have better accuracy than by VS. But with the noises increasing, IESIL's performance will get worse due to the reduction of the formatching sequences, which is caused by the crossinglost between the isograms and the intersection lines. VS keeps more stable both in positioning and orientating, although the results are not very satisfactory.
As indicated in Figure 12(c) IESIL is better than iICP on runtime, although it is worse than VS. It can be concluded that IESIL is a good compromise scheme, which balances both positioning accuracy and runtime. All in all, the combination algorithm is a better navigation technique compared to the traditional correlationtype algorithms and fits for the specialties of UMN.
5.2. Further Improvements

(A)
The recomparison in Step 4 of IESIL can be expanded to the four vertexes of the subcell, which is determined by quartering the grid cell of previous step and takes the EEpoint (solved in Step 3) on the first contour as center. Then the same recomparison process is carried out iteratively, and this can increase the algorithm's faulttolerance.

(B)
IESIL may encounter the failure of searching the intersection lines in case of large measurement errors. If IESIL finds no matching points finally, VS, which is relatively more robust under big errors shall be assumed as the alternative. Actually, IESIL, VS and iICP shall be programmed in parallel for crossreferencing.

(C)
IESIL's performance will decrease with noises increasing, so some good noisesreducing methods are necessary. Additionally, the great discrete degree of samplings occasionally existed will increase the difficulty of noises recognition. Waveletbased noises removal methods accompanied with multiscale analysis can be applied on the measurement sequence to ensure the new algorithm producing better results.

(D)
The geomagnetic gradient measurement can reduce the influences of the daily variation and reflect the real distribution of the geomagnetic field more accurately. Hausdorffbased RC with gradients as variables will have higher accuracy.
6. Conclusions
This paper proposes an appropriate matching scheme and a combined positioning algorithm for UMN. Hausdorffbased RC preliminarily gives a solution for the most confusing issue caused by the geomagnetic fluctuations, such as daily variation. IESIL overcomes not only VS's inherent low accuracy caused by the assumption ofno heading error but also iICP's timeconsuming which sources from the iterative rotation and shifting operations. The experiments based on the simulation platform and the realsurveyed data both validate the new algorithm, and the efficiency and accuracy of the combination algorithm are also relatively satisfactory. In view that the research of UMN is just in the beginning phase, the aforementioned work has given a basic verification of UMN's potentiality.
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Acknowledgment
This work is supported by the National High Technology Research and Development Program of China Grant no. 2007AA09Z201.
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Lin, Y. HausdorffBased RC and IESIL Combined Positioning Algorithm for Underwater Geomagnetic Navigation. EURASIP J. Adv. Signal Process. 2010, 593238 (2010). https://doi.org/10.1155/2010/593238
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DOI: https://doi.org/10.1155/2010/593238